%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: Form_Gal.dvi %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips Form_Gal %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2001.06.04:1747 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 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Ft(H)21 b Fu(is)16 b(giv)o(en)f(b)o(y)h(the) g(comm)o(utativ)n(e)d(diagram)670 764 y(~)657 777 y Ft(H)t Fu(\()p Ft(A)p Fu(\))e Fr(\002)849 764 y Fu(~)837 777 y Ft(H)t Fu(\()p Ft(A)p Fu(\))1150 764 y(~)1137 777 y Ft(H)t Fu(\()p Ft(A)p Fu(\))p 970 761 153 2 v 1081 760 a Fk(-)626 967 y Ft(k)r Fu(-)p Fr(A)p Fq(lg)q Fu(\()p Ft(H)k Fr(\012)c Ft(H)q(;)d(A)p Fu(\))81 b Ft(k)r Fu(-)q Fr(A)p Fq(lg)p Fu(\()p Ft(H)q(;)8 b(A)p Fu(\))p 1001 956 53 2 v 1012 955 a Fk(-)p 806 921 2 127 v 806 921 a(?)p 1195 921 V 348 w(?)278 1049 y Fu(So)19 b(the)f(group-ring)j Ft(k)r Fn(Z)e Fu(has)g(b)q(een)g(seen)f(to)i(b)q(e)f(a)g(Hopf)g (algebra)g(with)f(the)228 1107 y(diagonal)f(\001\()p Ft(g)r Fu(\))d(=)h Ft(g)e Fr(\012)e Ft(g)19 b Fu(for)e(ev)o(ery)e (elemen)o(t)f Ft(g)k Fu(in)f(the)f(group)i Fn(Z)p Fu(.)k(This)17 b(holds)228 1165 y(not)24 b(only)g(for)h(the)f(group)h Fn(Z)p Fu(.)45 b(Ev)o(ery)23 b(group-ring)i Ft(k)r(G)g Fu(is)f(a)h(Hopf)f(algebra)228 1223 y(with)19 b(the)g(same)f(com)o (ultipli)o(cation,)f(ev)o(en)h(for)h(non-comm)o(utativ)o(e)d(groups)k Ft(G)p Fu(.)228 1281 y(The)15 b(non-comm)o(utativ)o(e)d(group)j(rings,) g(ho)o(w)o(ev)o(er,)f(do)h(not)h(an)o(ymore)d(represen)o(t)228 1339 y(group)19 b(v)m(alued)f(functors)h(on)g Ft(k)r Fu(-)p Fr(A)p Fq(lg)q Fu(.)28 b(They)18 b(are)g(sp)q(ecial)g(instances) h(of)f(formal)228 1397 y(groups.)278 1455 y(Another)e(concrete)f (example)f(of)j(a)f(group)h(v)m(alued)g(functor)f(is)793 1543 y Ft(C)9 b Fu(:)16 b Ft(k)r Fu(-)p Fr(A)p Fq(lg)f Fr(\000)-31 b(!)14 b(G)m Fq(rp)p Ft(;)228 1632 y Fu(the)21 b(circle)e(group,)k(de\014ned)e(b)o(y)f Ft(C)t Fu(\()p Ft(A)p Fu(\))i(:=)f Fr(f)p Fu(\()p Ft(a;)8 b(b)p Fu(\))22 b Fr(2)g Ft(A)14 b Fr(\002)g Ft(A)p Fr(j)p Ft(a)1451 1613 y Fo(2)1484 1632 y Fu(+)h Ft(b)1558 1613 y Fo(2)1599 1632 y Fu(=)22 b(1)p Fr(g)p Fu(.)228 1690 y(The)g(group)i(structure)e (is)g(giv)o(en)g(b)o(y)g(\()p Ft(a;)8 b(b)p Fu(\))15 b Fr(\003)g Fu(\()p Ft(c;)8 b(d)p Fu(\))24 b(:=)g(\()p Ft(ac)15 b Fr(\000)g Ft(bd;)8 b(ad)16 b Fu(+)f Ft(bc)p Fu(\).)228 1748 y(The)24 b(represen)o(ting)f(Hopf)h(algebra)h(is)f(the) g("trigonometric)f(algebra")i Ft(H)31 b Fu(=)228 1806 y Ft(k)r Fu([)p Ft(c;)8 b(s)p Fu(])p Ft(=)p Fu(\()p Ft(c)413 1788 y Fo(2)443 1806 y Fu(+)j Ft(s)515 1788 y Fo(2)546 1806 y Fr(\000)f Fu(1\).)22 b(The)16 b(diagonal)h(is)f(de\014ned)g(b)o (y)724 1893 y(\001\()p Ft(c)p Fu(\))41 b(=)g Ft(c)11 b Fr(\012)g Ft(c)g Fr(\000)g Ft(s)g Fr(\012)g Ft(s;)722 1951 y Fu(\001\()p Ft(s)p Fu(\))41 b(=)g Ft(c)11 b Fr(\012)g Ft(s)g Fu(+)g Ft(s)g Fr(\012)g Ft(c:)278 2037 y Fu(The)j(most)g(in)o (teresting)f(observ)m(ation)i(is)f(this.)21 b(Let)14 b Ft(A)g Fu(b)q(e)h(a)f(comm)o(utativ)o(e)d Ft(k)r Fu(-)228 2096 y(algebra)g(with)g(2)g(in)o(v)o(ertible)e(and)i(con)o(taining)g Ft(i)j Fu(=)1148 2056 y Fr(p)p 1190 2056 64 2 v 40 x(\000)p Fu(1.)19 b(Then)12 b(the)e(assignmen)o(t)463 2212 y Ft(U)5 b Fu(\()p Ft(A)p Fu(\))13 b Fr(3)h Ft(a)g Fr(7!)739 2141 y Fh(\022)781 2178 y Fu(1)p 781 2200 25 2 v 781 2246 a(2)810 2212 y(\()p Ft(a)d Fu(+)g Ft(a)941 2191 y Fp(\000)p Fo(1)988 2212 y Fu(\))p Ft(;)1042 2178 y Fu(1)p 1034 2200 42 2 v 1034 2246 a(2)p Ft(i)1080 2212 y Fu(\()p Ft(a)f Fr(\000)h Ft(a)1211 2191 y Fp(\000)p Fo(1)1258 2212 y Fu(\))1277 2141 y Fh(\023)1327 2212 y Fr(2)j Ft(C)t Fu(\()p Ft(A)p Fu(\))228 2333 y(de\014nes)19 b(a)h(functorial)f (isomorphism)e(of)j(groups.)32 b(If)19 b(2)1268 2315 y Fp(\000)p Fo(1)1315 2333 y Ft(;)8 b(i)19 b Fr(2)g Ft(k)j Fu(then)d Ft(U)25 b Fu(and)228 2391 y Ft(C)e Fu(are)c(isomorphic)f (group)j(v)m(alued)e(functors,)h(hence)f(they)g(ha)o(v)o(e)g (isomorphic)228 2449 y(represen)o(ting)c(Hopf)h(algebras)640 2537 y Ft(k)r Fu([)p Ft(x;)8 b(x)759 2516 y Fp(\000)p Fo(1)805 2537 y Fu(])833 2523 y Fr(\030)833 2539 y Fu(=)885 2537 y Ft(k)r Fu([)p Ft(c;)g(s)p Fu(])p Ft(=)p Fu(\()p Ft(c)1070 2516 y Fo(2)1100 2537 y Fu(+)j Ft(s)1172 2516 y Fo(2)1203 2537 y Fr(\000)g Fu(1\))p Ft(:)p eop %%Page: 4 4 4 3 bop 228 55 a Fl(4)564 b(BODO)13 b(P)m(AREIGIS)228 154 y Fu(If)h Ft(i)19 b(=)-30 b Fr(2)14 b Ft(k)j Fu(then)d(the)g(t)o(w) o(o)g(group)i(v)m(alued)e(functors)h(are)f(not)h(isomorphic,)e(neither) 228 212 y(are)j(their)f(represen)o(ting)g(Hopf)i(algebras)f Ft(k)r Fu([)p Ft(x;)8 b(x)1128 194 y Fp(\000)p Fo(1)1174 212 y Fu(])16 b(and)h Ft(k)r Fu([)p Ft(c;)8 b(s)p Fu(])p Ft(=)p Fu(\()p Ft(c)1484 194 y Fo(2)1514 212 y Fu(+)i Ft(s)1585 194 y Fo(2)1616 212 y Fr(\000)g Fu(1\).)278 270 y(If)20 b Ft(k)k Fu(is)d(a)h(\014eld)e(of)i(c)o(haracterictic)d Fr(6)p Fu(=)j(2)g(and)g Ft(i)28 b(=)-30 b Fr(2)23 b Ft(k)r Fu(,)f(then)f Ft(U)27 b Fu(and)21 b Ft(C)k Fu(are)228 329 y(non-isomorphic)13 b(but)i(they)f(induce)f(isomorphic)g(functors)h Ft(U)5 b Fr(j)1398 336 y Fi(k)q Fo(\()p Fi(i)p Fo(\))1474 329 y Fu(and)14 b Ft(C)t Fr(j)1619 336 y Fi(k)q Fo(\()p Fi(i)p Fo(\))1694 329 y Fu(if)228 387 y(restricted)f(to)h(the)g Ft(k)r Fu(\()p Ft(i)p Fu(\)-algebras.)21 b(Let)14 b Ft(K)k Fu(=)13 b Ft(k)r Fu(\()p Ft(i)p Fu(\))h(and)h(let)e Ft(A)g Fu(b)q(e)h(a)h Ft(K)t Fu(-algebra.)228 445 y(Then)h(w)o(e)g(ha)o(v)o(e) 295 510 y Ft(K)t Fu(-)p Fr(A)p Fq(lg)q Fu(\()p Ft(K)f Fr(\012)c Ft(k)r Fu([)p Ft(x;)d(x)675 492 y Fp(\000)p Fo(1)721 510 y Fu(])p Ft(;)g(A)p Fu(\))853 496 y Fr(\030)854 512 y Fu(=)906 510 y Ft(k)r Fu(-)p Fr(A)p Fq(lg)q Fu(\()p Ft(k)r Fu([)p Ft(x;)g(x)1162 492 y Fp(\000)p Fo(1)1208 510 y Fu(])p Ft(;)g(A)p Fu(\))853 554 y Fr(\030)854 570 y Fu(=)906 568 y Ft(U)d Fr(j)958 575 y Fi(K)992 568 y Fu(\()p Ft(A)p Fu(\))853 613 y Fr(\030)854 629 y Fu(=)906 626 y Ft(C)t Fr(j)959 633 y Fi(K)992 626 y Fu(\()p Ft(A)p Fu(\))853 671 y Fr(\030)854 687 y Fu(=)906 685 y Ft(k)r Fu(-)p Fr(A)p Fq(lg)q Fu(\()p Ft(k)r Fu([)p Ft(c;)j(s)p Fu(])p Ft(=)p Fu(\()p Ft(c)1228 666 y Fo(2)1258 685 y Fu(+)j Ft(s)1330 666 y Fo(2)1361 685 y Fr(\000)f Fu(1\))p Ft(;)e(A)p Fu(\))853 729 y Fr(\030)854 745 y Fu(=)906 743 y Ft(K)t Fu(-)p Fr(A)p Fq(lg)q Fu(\()p Ft(K)15 b Fr(\012)c Ft(k)r Fu([)p Ft(c;)d(s)p Fu(])p Ft(=)p Fu(\()p Ft(c)1352 725 y Fo(2)1382 743 y Fu(+)j Ft(s)1454 725 y Fo(2)1485 743 y Fr(\000)f Fu(1\))p Ft(;)e(A)p Fu(\))228 824 y(hence)19 b Ft(K)f Fr(\012)13 b Ft(k)r Fu([)p Ft(x;)8 b(x)597 806 y Fp(\000)p Fo(1)643 824 y Fu(])677 810 y Fr(\030)677 826 y Fu(=)735 824 y Ft(K)18 b Fr(\012)13 b Ft(k)r Fu([)p Ft(c;)8 b(s)p Fu(])p Ft(=)p Fu(\()p Ft(c)1031 806 y Fo(2)1064 824 y Fu(+)13 b Ft(s)1138 806 y Fo(2)1172 824 y Fr(\000)g Fu(1\))20 b(as)h Ft(K)t Fu(-Hopf)f(algebras,)228 882 y(where)13 b(the)h(tensor)g(pro)q(duct)g(is)g(alw)o(a)o(ys)f(tak)o (en)h(o)o(v)o(er)e(the)i(base)g(ring)g Ft(k)r Fu(.)20 b(Observ)o(e)228 941 y(that)c(a)h(cancellation)e(prop)q(ert)o(y)i (cannot)f(b)q(e)h(exp)q(ected)e(in)h(this)g(case.)278 999 y(In)d(particular,)h(the)f Fn(Q)p Fu(-Hopf)h(algebras)h Fn(Q)p Fu([)p Ft(x;)8 b(x)1151 981 y Fp(\000)p Fo(1)1197 999 y Fu(])13 b(and)h Fn(Q)p Fu([)p Ft(c;)8 b(s)p Fu(])p Ft(=)p Fu(\()p Ft(c)1516 981 y Fo(2)1541 999 y Fu(+)e Ft(s)1608 981 y Fo(2)1634 999 y Fr(\000)g Fu(1\))228 1057 y(and)21 b(the)g Fn(R)p Fu(-Hopf)g(algebras)h Fn(R)p Fu([)p Ft(x;)8 b(x)927 1039 y Fp(\000)p Fo(1)972 1057 y Fu(])21 b(and)g Fn(R)p Fu([)p Ft(c;)8 b(s)p Fu(])p Ft(=)p Fu(\()p Ft(c)1306 1039 y Fo(2)1339 1057 y Fu(+)15 b Ft(s)1415 1039 y Fo(2)1448 1057 y Fr(\000)f Fu(1\))22 b(are)f(not)228 1115 y(isomorphic,)16 b(but)h(the)g Fn(C)p Fu(-Hopf)h(algebras)g Fn(C)p Fu([)p Ft(x;)8 b(x)1165 1097 y Fp(\000)p Fo(1)1211 1115 y Fu(])1241 1101 y Fr(\030)1241 1117 y Fu(=)1295 1115 y Fn(C)p Fu([)p Ft(c;)g(s)p Fu(])p Ft(=)p Fu(\()p Ft(c)1493 1097 y Fo(2)1524 1115 y Fu(+)k Ft(s)1597 1097 y Fo(2)1628 1115 y Fr(\000)g Fu(1\))228 1173 y(are.)21 b(This)16 b(is)g(an)h(example)d(for)j(the)f(next)g (de\014nition.)228 1241 y Fn(De\014nition)g(1.2.)i Fu(Let)d Ft(G)h Fu(and)f Ft(G)865 1223 y Fp(0)892 1241 y Fu(b)q(e)g(group)g(v)m (alued)g(functors)g(on)g Ft(k)r Fu(-)q Fr(A)p Fq(lg)p Fu(.)21 b(Let)228 1300 y Ft(K)26 b Fu(b)q(e)d(a)g(faithfully)f(\015at)h (comm)o(utativ)n(e)c Ft(k)r Fu(-algebra.)41 b(If)23 b(the)f (restrictions)g(to)228 1358 y Ft(K)t Fu(-)p Fr(A)p Fq(lg)16 b Fu(are)f(isomorphic)f(group)i(v)m(alued)e(functors:)21 b Ft(G)p Fr(j)1252 1365 y Fi(K)1301 1344 y Fr(\030)1301 1360 y Fu(=)1353 1358 y Ft(G)1391 1340 y Fp(0)1403 1358 y Fr(j)1417 1365 y Fi(K)1451 1358 y Fu(,)15 b(then)g Ft(G)h Fu(and)228 1416 y Ft(G)266 1398 y Fp(0)294 1416 y Fu(are)g(called)g Ft(K)t Fq(-forms)g Fu(of)g(eac)o(h)g(other)g(as)h (groups.)278 1474 y(Let)i Ft(H)k Fu(and)d Ft(H)573 1456 y Fp(0)604 1474 y Fu(b)q(e)g(Hopf)f(algebras)g(o)o(v)o(er)g(the)g(comm) o(utativ)n(e)d(ring)j Ft(k)r Fu(.)30 b(Let)228 1532 y Ft(K)21 b Fu(b)q(e)d(a)f(faithfully)f(\015at)i(comm)o(utativ)n(e)c Ft(k)r Fu(-algebra.)26 b(If)16 b Ft(K)g Fr(\012)c Ft(H)1444 1518 y Fr(\030)1444 1534 y Fu(=)1498 1532 y Ft(K)k Fr(\012)11 b Ft(H)1649 1514 y Fp(0)1679 1532 y Fu(as)228 1590 y Ft(K)t Fu(-Hopf)18 b(algebras,)g(then)f Ft(H)22 b Fu(and)d Ft(H)930 1572 y Fp(0)959 1590 y Fu(are)f(called)e Ft(K)t Fq(-forms)i Fu(of)g(eac)o(h)f(other)h(as)228 1648 y(Hopf)e(algebras)278 1706 y(W)l(e)j(sa)o(y)i(that)f Ft(G)h Fu(and)g Ft(G)759 1688 y Fp(0)791 1706 y Fu(resp.)33 b Ft(H)24 b Fu(and)d Ft(H)1132 1688 y Fp(0)1164 1706 y Fu(are)f Fq(forms)f Fu(of)i(eac)o(h)e(other)i(if)228 1764 y(there)14 b(exists)g(a)h (faithfully)e(\015at)i Ft(k)r Fu(-algebra)h Ft(K)j Fu(suc)o(h)14 b(that)h(they)f(are)h Ft(K)t Fu(-forms)f(of)228 1823 y(eac)o(h)i(other.)278 1891 y(So)d(for)g Ft(G)g Fu(and)g Ft(G)593 1873 y Fp(0)618 1891 y Fu(to)g(b)q(e)g Ft(K)t Fu(-forms)f(of)h(eac)o(h)f(other)h(w)o(e)f(need)g(an)h(isomorphism)228 1949 y(of)j(set)g(v)m(alued)h(functors)f Ft(\013)6 b Fu(:)16 b Ft(G)p Fr(j)820 1956 y Fi(K)869 1949 y Fr(\000)-31 b(!)14 b Ft(G)979 1931 y Fp(0)991 1949 y Fr(j)1005 1956 y Fi(K)1055 1949 y Fu(suc)o(h)i(that)560 2042 y Ft(G)p Fr(j)612 2049 y Fi(K)657 2042 y Fr(\002)11 b Ft(G)p Fr(j)759 2049 y Fi(K)1133 2042 y Ft(G)1171 2024 y Fp(0)1183 2042 y Fr(j)1197 2049 y Fi(K)1242 2042 y Fr(\002)g Ft(G)1330 2024 y Fp(0)1342 2042 y Fr(j)1356 2049 y Fi(K)p 808 2030 312 2 v 1077 2029 a Fk(-)902 2015 y Ft(\013)g Fr(\002)g Ft(\013)633 2237 y(G)p Fr(j)685 2244 y Fi(K)1213 2237 y Ft(G)1251 2219 y Fp(0)1263 2237 y Fr(j)1277 2244 y Fi(K)p 734 2225 465 2 v 1157 2224 a Fk(-)951 2215 y Ft(\013)p 676 2190 2 127 v 677 2190 a Fk(?)p 1261 2190 V 543 w(?)278 2304 y Fu(comm)o(ute)o(s.)278 2363 y(There)g(ma)o(y)g(b)q(e)h(man)o(y)f (di\013eren)o(t)g(Hopf)i(algebras)f Ft(H)1240 2344 y Fp(0)1264 2363 y Fu(whic)o(h)g(are)g(forms)f(for)i Ft(H)228 2421 y Fu(with)h(resp)q(ect)f(to)h(some)f(faithfully)g(\015at)h (extension)g Ft(K)t Fu(.)20 b(In)14 b(particular)f(the)h(ric)o(h-)228 2479 y(ness)k(of)g(Hopf)g(algebras)h(o)o(v)o(er)e Fn(Q)h Fu(should)h(b)q(e)f(higher)g(than)g(o)o(v)o(er)f Fn(C)p Fu(.)27 b(Gran)o(ted)228 2537 y(there)16 b(ma)o(y)f(b)q(e)h(Hopf)h (algebras)g(de\014ned)g(o)o(v)o(er)e Fn(C)p Fu(,)i(whic)o(h)f(do)h(not) g(come)e(ab)q(out)p eop %%Page: 5 5 5 4 bop 465 55 a Fl(F)o(ORMS)18 b(OF)e(HOPF)h(ALGEBRAS)g(AND)g(GALOIS)f (THEOR)m(Y)218 b(5)228 154 y Fu(b)o(y)17 b(a)i(base)f(ring)g(extension) f(from)g 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y(isomorphic)16 b(to)i(the)f(automorphism)g(group)h Fr(G)s Fq(al)p Fu(-Aut\()p Ft(E)1313 2403 y Fi(F)1310 2434 y(k)1342 2421 y Fu(\))g(of)g(the)f(trivial)g Ft(F)7 b Fu(-)228 2479 y(Galois)19 b(extension)e Ft(E)636 2461 y Fi(F)633 2492 y(k)684 2479 y Fu(of)i Ft(k)r Fu(.)27 b(This)19 b(Galois)f(extension)g(can)g(b)q(e)h(describ)q(ed)f(b)o(y)228 2537 y(the)c(ring)h Ft(E)448 2519 y Fi(F)445 2550 y(k)491 2537 y Fu(=)f(\()p Ft(k)r(F)7 b Fu(\))647 2519 y Fp(\003)666 2537 y Fu(,)15 b(the)f(dual)h(space)g(of)g(the)g(group)h(ring)e Ft(k)r(F)7 b Fu(,)14 b(on)i(whic)o(h)e Ft(F)p eop %%Page: 6 6 6 5 bop 228 55 a Fl(6)564 b(BODO)13 b(P)m(AREIGIS)228 154 y Fu(acts)i(b)o(y)g(automorphisms)f(in)h(suc)o(h)f(a)i(w)o(a)o(y)l (,)f(that)g(the)g(ring)g(extension)g(\()p Ft(k)r(F)7 b Fu(\))1652 136 y Fp(\003)1671 154 y Ft(=k)228 212 y Fu(is)14 b(an)g Ft(F)7 b Fu(-Galois)14 b(extension)g(in)g(the)g(sense)g (of)g([CHR].)f(Actually)f(this)i(leads)g(to)h(a)228 270 y(functorial)d(isomorphism)f Fn(Aut)p Fu(\()p 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Fr(\000)p Ft(asc)p Fr(\000)p Ft(bc)734 1441 y Fo(2)752 1462 y Fu(+)p Ft(u;)g Fu(\()p Ft(c)p Fr(\000)p Fu(2\)\()p Ft(c)p Fr(\000)p Fu(1\)\()p Ft(c)p Fu(+1\)\()p Ft(c)p Fu(+2\))p Ft(;)g Fu(\()p Ft(c)p Fr(\000)p Fu(1\)\()p Ft(c)p Fu(+1\)\()p Ft(sc)p Fr(\000)p Fu(2)p Ft(a)p Fu(\)\))p Ft(:)278 1567 y Fq(In)18 b(al)r(l)h(c)n(ases)e Ft(a;)8 b(b;)g(u)14 b Fr(2)g Ft(k)20 b Fq(satisfy)d Ft(a)939 1549 y Fo(2)970 1567 y Fu(+)11 b(4)p Ft(b)k Fu(=)f Ft(u)j Fq(and)h Ft(u)g Fq(is)g(a)f(unit)i(in)f Ft(k)r Fq(.)23 b(The)228 1625 y(Hopf)17 b(algebr)n(a)h(structur)n(e)g(in)f(al)r(l)i(c) n(ases)f(is)f(de\014ne)n(d)h(by)378 1727 y Fu(\001\()p Ft(c)p Fu(\))41 b(=)14 b Ft(u)599 1709 y Fp(\000)p Fo(1)646 1727 y Fu(\(\()p Ft(a)710 1709 y Fo(2)740 1727 y Fu(+)d(2)p Ft(b)p Fu(\))p Ft(c)g Fr(\012)g Ft(c)g Fr(\000)g Ft(a)p Fu(\()p Ft(c)g Fr(\012)f Ft(s)h Fu(+)g Ft(s)g Fr(\012)g Ft(c)p Fu(\))g(+)g(2)p Ft(s)h Fr(\012)f Ft(s)p Fu(\))p Ft(;)376 1785 y Fu(\001\()p Ft(s)p Fu(\))41 b(=)14 b Ft(u)599 1767 y Fp(\000)p 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(Galois)g(extension)g(\()p Ft(k)r(C)1101 2186 y Fo(2)1120 2179 y Fu(\))1139 2160 y Fp(\003)1173 2165 y Fr(\030)1173 2181 y Fu(=)1225 2179 y Ft(k)t Fr(\002)s Ft(k)j Fu(of)d Ft(k)i Fu(as)f(will)e(b)q(e)i(seen)228 2237 y(b)q(elo)o(w.)33 b(Since)19 b(the)h(automorphism)f(groups)i Fn(Aut)p Fu(\()p Ft(k)r(G)p Fu(\))1323 2223 y Fr(\030)1324 2239 y Fu(=)1382 2237 y Fr(G)s Fq(al)p Fu(-)q Fn(Aut)p Fu(\()p Ft(k)16 b Fr(\002)d Ft(k)r Fu(\))228 2295 y(coincide,)24 b(the)f(\014rst)h (Amitsur)e(cohomology)h(groups)i(describing)e(the)g(forms)228 2353 y(coincide,)13 b(to)q(o.)22 b(So)14 b(there)g(is)g(a)h(bijectiv)o (e)d(corresp)q(ondence)i(b)q(et)o(w)o(een)g(the)g(forms)228 2411 y(of)d(the)g(group-rings)i(in)e(the)g(theorem)f(and)i(the)f (quadratic)g(Galois)h(extensions)f(of)228 2469 y Ft(k)17 b Fu([see)e(Thm.)20 b(4.1].)h(This)16 b(corresp)q(ondence)g(w)o(as)g (used)f(to)h(explicitly)d(calculate)228 2527 y(the)j(forms)f(giv)o(en)h (in)f(the)h(theorem.)p eop %%Page: 7 7 7 6 bop 465 55 a Fl(F)o(ORMS)18 b(OF)e(HOPF)h(ALGEBRAS)g(AND)g(GALOIS)f (THEOR)m(Y)218 b(7)634 154 y Fu(2.)43 b Fs(Hopf)17 b(Galois)h (extensions)278 241 y Fu(A)c(di\013eren)o(t)f(class)i(of)f("forms")h (is)f(obtained)h(if)e(one)i(considers)f(the)h(follo)o(wing)228 299 y(cancellation)g(problem.)228 373 y Fn(De\014nition)f(2.1.)j Fu(Let)c Ft(G)6 b Fu(:)16 b Ft(k)r Fu(-)p Fr(A)p Fq(lg)f Fr(\000)-30 b(!)13 b(G)m Fq(rp)g Fu(b)q(e)g(a)g(group)h(v)m(alued)f (functor.)20 b(Then)228 431 y(the)d(m)o(ultipli)o(cation)d(of)k Ft(G)f Fu(on)h(itself)e Ft(G)c Fr(\002)f Ft(G)16 b Fr(\000)-30 b(!)15 b Ft(G)i Fu(mak)o(es)f Ft(G)h Fu(a)h Ft(G)p Fu(-set)f(v)m(alued) 228 489 y(functor.)k(Here)15 b(w)o(e)g(de\014ne)h(the)f(functor)h Ft(G)11 b Fr(\002)f Ft(G)17 b Fu(b)o(y)e(\()p Ft(G)c Fr(\002)f Ft(G)p Fu(\)\()p Ft(K)t Fu(\))k(:=)g Ft(G)p Fu(\()p Ft(K)t Fu(\))c Fr(\002)228 547 y Ft(G)p Fu(\()p Ft(K)t Fu(\),)i(so)g(that)g(the)f(m)o(ultipication)e(of)i(eac)o(h)g (group)i Ft(G)p Fu(\()p Ft(K)t Fu(\))f(de\014nes)f(a)h(functorial)228 606 y(homomorphism)j Ft(G)e Fr(\002)f Ft(G)18 b Fr(\000)-31 b(!)17 b Ft(G)p Fu(,)i(brie\015y)e(the)h(m)o(ultiplication)d(on)k Ft(G)g Fu(and)f(the)228 664 y Ft(G)p Fu(-set)f(structure)f(is)g (de\014ned)g("comp)q(onen)o(t)o(wise".)278 722 y(Let)e Ft(X)c Fu(:)16 b Ft(k)r Fu(-)p Fr(A)p Fq(lg)f Fr(\000)-30 b(!)13 b(S)n Fq(et)j Fu(b)q(e)e(another)h(functor)f(whic)o(h)g(is)g (also)h(a)f Ft(G)p Fu(-set)h(v)m(alued)228 780 y(functor)i(b)o(y)g Ft(X)g Fr(\002)11 b Ft(G)17 b Fr(\000)-31 b(!)16 b Ft(X)t Fu(.)25 b(Let)18 b Ft(K)j Fu(b)q(e)d(a)g(faithfully)e(\015at)i(comm)o (utativ)n(e)c(ring)228 838 y(extension)d(of)g Ft(k)r Fu(.)20 b(If)11 b(the)g(restrictions)g Ft(G)p Fr(j)973 845 y Fi(K)1019 838 y Fu(and)h Ft(X)t Fr(j)1167 845 y Fi(K)1213 838 y Fu(to)g Ft(K)t Fu(-)p Fr(A)p Fq(lg)g Fu(are)g(isomorphic)228 896 y(as)18 b Ft(G)p Fr(j)341 903 y Fi(K)375 896 y Fu(-set)f(v)m(alued)h(functors,)f(then)g Ft(G)h Fu(and)f Ft(X)22 b Fu(are)17 b(called)f Ft(K)t Fq(-forms)h Fu(of)h(eac)o(h)228 954 y(other)e(as)h Ft(G)p Fu(-set)g(v)m(alued)f(functors.)278 1028 y(So)d(for)h Ft(G)f Fu(and)h Ft(X)j Fu(to)d(b)q(e)f Ft(K)t Fu(-forms)f(of)i(eac)o(h) e(other)i(w)o(e)e(need)h(an)g(isomorphism)228 1086 y(of)j(set)g(v)m (alued)h(functors)f Ft(\013)6 b Fu(:)16 b Ft(G)p Fr(j)820 1093 y Fi(K)869 1086 y Fr(\000)-31 b(!)14 b Ft(X)t Fr(j)999 1093 y Fi(K)1050 1086 y Fu(suc)o(h)i(that)662 1192 y Ft(G)p Fr(j)714 1199 y Fi(K)759 1192 y Fr(\002)11 b Ft(G)p Fr(j)861 1199 y Fi(K)1049 1192 y Ft(X)t Fr(j)1107 1199 y Fi(K)1152 1192 y Fr(\002)g Ft(G)p Fr(j)1254 1199 y Fi(K)p 910 1180 125 2 v 993 1179 a Fk(-)914 1166 y Ft(\013)g Fr(\002)g Fu(1)735 1387 y Ft(G)p Fr(j)787 1394 y Fi(K)1122 1387 y Ft(X)t Fr(j)1180 1394 y Fi(K)p 836 1375 273 2 v 1067 1374 a Fk(-)956 1365 y Ft(\013)p 778 1340 2 127 v 779 1340 a Fk(?)p 1168 1340 V 348 w(?)278 1465 y Fu(comm)o(ute)o(s.) 278 1523 y(A)k(Hopf)h(algebraic)g(description)g(of)g(this)g(is)g (somewhat)g(more)f(complicated.)228 1581 y(The)h(notion)h(of)g(a)g Ft(G)p Fu(-set)h(and)f(of)g(forms)e(of)i(a)g Ft(G)p Fu(-set)g (translated)g(to)g(the)g(repre-)228 1639 y(sen)o(ting)d(ob)s(jects)g (of)h(the)f(represen)o(table)f(functors)i Ft(G)g Fu(and)g Ft(X)k Fu(giv)o(es)14 b(the)g(follo)o(w-)228 1697 y(ing)i (de\014nition.)228 1771 y Fn(De\014nition)k(2.2.)i Fu(Let)d Ft(H)736 1753 y Fp(\003)775 1771 y Fu(b)q(e)g(a)g(comm)o(utativ)n(e)d (Hopf)j(algebra)g(and)g Ft(A)g Fu(b)q(e)g(a)228 1829 y(comm)o(utativ)n(e)11 b(algebra.)21 b Ft(A)14 b Fu(is)g(called)f(an)i Ft(H)1050 1811 y Fp(\003)1070 1829 y Fq(-c)n(omo)n(dule)h(algebr)n(a)f Fu(if)f(there)g(is)g(an)228 1887 y(algebra)j(map)e Ft(\037)6 b Fu(:)15 b Ft(A)f Fr(\000)-31 b(!)14 b Ft(A)d Fr(\012)f Ft(H)837 1869 y Fp(\003)874 1887 y Fu(suc)o(h)16 b(that)g(the)g (diagrams)312 2035 y Ft(A)291 b(A)10 b Fr(\012)h Ft(H)781 2017 y Fp(\003)p 363 2021 263 2 v 584 2020 a Fk(-)479 2001 y Ft(\037)250 2230 y(A)f Fr(\012)h Ft(H)391 2212 y Fp(\003)577 2230 y Ft(A)g Fr(\012)g Ft(H)719 2212 y Fp(\003)750 2230 y Fr(\012)g Ft(H)844 2212 y Fp(\003)p 426 2216 138 2 v 522 2215 a Fk(-)436 2196 y Ft(\037)g Fr(\012)g Fu(1)p 330 2181 2 127 v 331 2181 a Fk(?)289 2122 y Fi(\037)p 720 2181 V 721 2181 a Fk(?)740 2128 y Fo(1)p Fp(\012)p Fo(\001)977 2215 y Fu(and)1211 2035 y Ft(A)291 b(A)10 b Fr(\012)h Ft(H)1680 2017 y Fp(\003)p 1262 2021 263 2 v 1483 2020 a Fk(-)1378 2001 y Ft(\037)1211 2231 y(A)309 b(A)11 b Fr(\012)g Ft(k)p 1262 2216 281 2 v 1501 2215 a Fk(-)1383 2206 y Fr(\030)p 1229 2181 2 127 v 1229 2181 a Fk(?)1169 2135 y Fu(id)p 1619 2181 V 1619 2181 a Fk(?)1639 2127 y Fo(1)p Fp(\012)p Fi(\017)278 2304 y Fu(comm)o(ute)o(.)278 2363 y(Let)21 b Ft(H)414 2344 y Fp(\003)456 2363 y Fu(b)q(e)h(a)g(comm)o(utativ)n(e)c(Hopf)k (algebra)g(and)g Ft(A)f Fu(b)q(e)h(a)g(comm)o(utativ)o(e)228 2421 y Ft(H)272 2403 y Fp(\003)292 2421 y Fu(-como)q(dule)g(algebra.)43 b(Let)24 b Ft(K)j Fu(b)q(e)d(a)g(faithfully)e(\015at)i(comm)o(utativ)n (e)c(ring)228 2479 y(extension)13 b(of)h Ft(k)r Fu(.)21 b(If)13 b Ft(K)e Fr(\012)6 b Ft(H)743 2461 y Fp(\003)776 2465 y Fr(\030)777 2481 y Fu(=)829 2479 y Ft(K)k Fr(\012)c Ft(A)14 b Fu(as)g Ft(K)d Fr(\012)6 b Ft(H)1174 2461 y Fp(\003)1194 2479 y Fu(-como)q(dule)13 b(algebras,)h(then)228 2537 y Ft(A)i Fu(is)g(called)f(a)i Ft(K)t Fq(-form)f Fu(of)g Ft(H)787 2519 y Fp(\003)807 2537 y Fu(.)p eop %%Page: 8 8 8 7 bop 228 55 a Fl(8)564 b(BODO)13 b(P)m(AREIGIS)278 154 y Fu(Closely)j(connected)h(with)g Ft(K)t Fu(-forms)g(of)h Ft(G)p Fu(-set)g(v)m(alued)f(functors)h(is)f(the)g(no-)228 212 y(tion)f(of)h(a)f(principal)f(homogeneous)i(space.)228 282 y Fn(De\014nition)23 b(2.3.)g Fu(If)e Ft(G)h Fu(is)f(a)h(group)g (and)g Ft(X)k Fu(is)21 b(a)h(set,)g(then)g(a)f Ft(G)p Fu(-set)i Ft(X)i Fu(is)228 340 y(called)c Fq(homo)n(gene)n(ous)p Fu(,)i(if)f(for)g(eac)o(h)g(pair)g Ft(x;)8 b(y)25 b Fr(2)f Ft(X)j Fu(there)21 b(exists)h(a)h Ft(g)j Fr(2)e Ft(G)228 399 y Fu(suc)o(h)16 b(that)h Ft(xg)f Fu(=)e Ft(y)r Fu(.)21 b(A)16 b Ft(G)p Fu(-set)h Ft(X)k Fu(is)16 b(a)h Fq(princip)n(al)g(homo) n(gene)n(ous)g Ft(G)p Fu(-set)g(if)f Ft(X)k Fu(is)228 457 y(homogeneous)c(and)h Ft(xg)f Fu(=)d Ft(x)j Fu(for)h(an)o(y)f Ft(x)e Fr(2)g Ft(X)20 b Fu(implies)14 b Ft(g)i Fu(=)e Ft(e)p Fu(.)278 527 y(It)g(is)i(easy)f(to)h(v)o(erify)l(,)d(that)i(a)h Ft(G)p Fu(-set)g Ft(X)k Fu(is)15 b(a)g(principal)g(homogeneous)g(space) 228 585 y(i\013)k(the)f(map)h Ft(')6 b Fu(:)16 b Ft(X)h Fr(\002)c Ft(G)19 b Fr(3)g Fu(\()p Ft(x;)8 b(g)r Fu(\))18 b Fr(7!)g Fu(\()p Ft(x;)8 b(xg)r Fu(\))18 b Fr(2)g Ft(X)g Fr(\002)12 b Ft(X)24 b Fu(is)18 b(bijectiv)o(e.)27 b(This)228 643 y(holds)14 b(also)f(in)h(the)f(case)g Ft(X)18 b Fu(=)c Fr(;)p Fu(.)20 b(If)13 b Ft(X)18 b Fr(6)p Fu(=)c Fr(;)f Fu(then)h Ft(X)j Fu(and)d Ft(G)g Fu(are)g(isomorphic)e(as)228 701 y Ft(G)p Fu(-sets.)22 b(These)16 b(statemen)o(ts)f(are)h(easily)g (translated)g(in)o(to)g(terms)f(of)h(functors.)278 759 y(The)f(map)f Ft(')6 b Fu(:)16 b Ft(X)d Fr(\002)c Ft(G)14 b Fr(\000)-30 b(!)13 b Ft(X)h Fr(\002)8 b Ft(X)20 b Fu(whic)o(h)14 b(is)h(de\014ned)g(for)h(an)o(y)f Ft(G)p Fu(-set)g(v)m(alued)228 817 y(functor)k Ft(X)24 b Fu(induces)19 b(the)h(algebra)g(homomorphism) c Ft( )21 b Fu(:)e Ft(A)13 b Fr(\012)g Ft(A)19 b Fr(3)h Ft(s)13 b Fr(\012)g Ft(t)19 b Fr(7!)228 838 y Fh(P)289 875 y Ft(st)330 883 y Fo(\()p Fi(A)p Fo(\))397 875 y Fr(\012)11 b Ft(t)465 883 y Fo(\()p Fi(H)511 874 y Fg(\003)527 883 y Fo(\))558 875 y Fr(2)j Ft(A)d Fr(\012)g Ft(H)747 857 y Fp(\003)784 875 y Fu(on)17 b(the)f(represen)o(ting)g(ob)s(jects.) 22 b Ft(')16 b Fu(is)h(an)g(isomor-)228 933 y(phism)e(i\013)h Ft( )i Fu(is.)228 1003 y Fn(Prop)r(osition)13 b(2.4.)k Fq(L)n(et)c Ft(G)h Fq(b)n(e)h(a)e(r)n(epr)n(esentable)i(gr)n(oup)e (value)n(d)i(functor)f(and)g Ft(X)228 1062 y Fq(b)n(e)k(a)f(r)n(epr)n (esentable)i Ft(G)p Fq(-set)g(value)n(d)g(functor)f(on)g Ft(k)r Fq(-)p Fr(A)p Fq(lg)q(.)23 b(L)n(et)17 b(the)i(r)n(epr)n (esenting)228 1120 y(algebr)n(a)g Ft(A)e Fq(of)h Ft(X)23 b Fq(b)n(e)18 b(faithful)r(ly)i(\015at.)k(Then)19 b Ft(G)g Fq(and)f Ft(X)23 b Fq(ar)n(e)17 b Ft(K)t Fq(-forms)h(of)g(e)n(ach)228 1178 y(other)f(as)g Ft(G)p Fq(-sets)h(for)e(some)h(faithful)r(ly)h (\015at)f(c)n(ommutative)h Ft(k)r Fq(-algebr)n(a)g Ft(K)j Fq(i\013)c Ft(X)228 1236 y Fq(is)g(a)g(princip)n(al)h(homo)n(gene)n (ous)f(sp)n(ac)n(e)g(over)h Ft(G)p Fq(.)228 1323 y(Pr)n(o)n(of.)h Fu(W)l(e)g(\014rst)h(remark)e(the)h(follo)o(wing.)31 b(Let)20 b Ft(X)k Fu(and)c Ft(Y)31 b Fu(b)q(e)19 b(represen)o(table)228 1381 y(functors,)e(let)f Ft(f)11 b Fu(:)16 b Ft(X)k Fr(\000)-31 b(!)15 b Ft(Y)28 b Fu(b)q(e)17 b(a)g(natural)h(transformation,)e(and)i Ft(K)j Fu(b)q(e)c(faith-)228 1439 y(fully)i(\015at.)34 b(Assume)19 b(that)h Ft(f)5 b Fr(j)798 1446 y Fi(K)838 1439 y Fu(:)18 b Ft(X)t Fr(j)928 1446 y Fi(K)983 1439 y Fr(\000)-30 b(!)20 b Ft(Y)12 b Fr(j)1116 1446 y Fi(K)1170 1439 y Fu(is)20 b(an)h(isomorphism.)31 b(Then)228 1498 y Ft(f)25 b Fu(is)19 b(an)i(isomorphism.)29 b(This)20 b(is)g(due)f(to)h(the)g(fact)g(that)g(the)f(corresp)q(onding)228 1556 y(statemen)o(t)14 b(holds)j(for)f(the)g(represen)o(ting)g (algebras.)278 1614 y(No)o(w)h(let)g(there)h(b)q(e)f(a)i(natural)f (isomorphism)e(of)i Ft(G)p Fr(j)1267 1621 y Fi(K)1301 1614 y Fu(-set)g(v)m(alued)g(functors)228 1672 y Ft(\013)6 b Fu(:)17 b Ft(G)p Fr(j)348 1679 y Fi(K)402 1672 y Fr(\000)-31 b(!)19 b Ft(X)t Fr(j)537 1679 y Fi(K)572 1672 y Fu(.)31 b(Then)19 b(since)g(\()p Ft(X)f Fr(\002)12 b Ft(Y)g Fu(\))p Fr(j)1071 1679 y Fi(K)1124 1672 y Fu(=)19 b Ft(X)t Fr(j)1239 1679 y Fi(K)1287 1672 y Fr(\002)13 b Ft(Y)e Fr(j)1392 1679 y Fi(K)1445 1672 y Fu(the)20 b(follo)o(wing)228 1730 y(diagram)15 b(comm)o(utes)652 1868 y(\()p Ft(G)c Fr(\002)g Ft(G)p Fu(\))p Fr(j)841 1875 y Fi(K)1042 1868 y Fu(\()p Ft(G)g Fr(\002)g Ft(G)p Fu(\))p Fr(j)1231 1875 y Fi(K)p 890 1856 138 2 v 986 1855 a Fk(-)919 1833 y Ft(')p Fr(j)965 1840 y Fi(K)649 2063 y Fu(\()p Ft(X)k Fr(\002)c Ft(G)p Fu(\))p Fr(j)844 2070 y Fi(K)1036 2063 y Fu(\()p Ft(X)k Fr(\002)c Ft(X)t Fu(\))p Fr(j)1237 2070 y Fi(K)p 893 2051 129 2 v 980 2050 a Fk(-)917 2028 y Ft(')p Fr(j)963 2035 y Fi(K)p 763 2016 2 127 v 764 2016 a Fk(?)676 1962 y Fi(\013)p Fp(\002)p Fo(1)p 1153 2016 V 1154 2016 a Fk(?)1173 1961 y Fi(\013)p Fp(\002)p Fi(\013)1287 2050 y Ft(:)278 2130 y Fu(Since)18 b Ft(G)p Fr(j)460 2137 y Fi(K)514 2130 y Fu(is)i(a)g(principal)e(homogeneous)i(space)f(o) o(v)o(er)g Ft(G)p Fr(j)1412 2137 y Fi(K)1466 2130 y Fu("comp)q(onen-) 228 2188 y(t)o(wise",)f(the)h(top)g(morphism)d(is)i(an)h(isomorphism.) 26 b(So)20 b(are)e(the)h(t)o(w)o(o)f(v)o(ertical)228 2246 y(arro)o(ws.)43 b(Th)o(us)24 b(the)f(b)q(ottom)g(arro)o(w)h(is)f (an)h(isomorphism.)40 b(By)23 b(the)g(ab)q(o)o(v)o(e)228 2304 y(argumen)o(t)15 b(w)o(e)h(get)g(that)h Ft(')6 b Fu(:)15 b Ft(X)h Fr(\002)11 b Ft(G)j Fr(\000)-30 b(!)13 b Ft(X)j Fr(\002)11 b Ft(X)20 b Fu(is)c(an)h(isomorphism.)278 2363 y(Con)o(v)o(ersely)f(if)h Ft(')6 b Fu(:)16 b Ft(X)g Fr(\002)c Ft(G)17 b Fr(\000)-31 b(!)16 b Ft(X)g Fr(\002)c Ft(X)22 b Fu(is)c(an)g(isomorphism,)d(then)i(in)g(par-)228 2421 y(ticular)22 b(the)h(induced)f(k-algebra)h(homomorphism)d Ft( )7 b Fu(:)19 b Ft(A)c Fr(\012)g Ft(A)25 b Fr(3)h Ft(s)15 b Fr(\012)h Ft(t)24 b Fr(7!)228 2441 y Fh(P)289 2479 y Ft(st)330 2487 y Fo(\()p Fi(A)p Fo(\))399 2479 y Fr(\012)15 b Ft(t)471 2487 y Fo(\()p Fi(H)517 2477 y Fg(\003)533 2487 y Fo(\))571 2479 y Fr(2)22 b Ft(A)14 b Fr(\012)g Ft(H)774 2461 y Fp(\003)815 2479 y Fu(of)21 b(the)g(represen)o(ting)f(algebras)i(is)e(an)i(isomor-)228 2537 y(phism.)i(\(Here)16 b(w)o(e)i(use)f(the)h(Sw)o(eedler)e(notation) j(in)e(con)o(text)g(with)g(a)h(bilinear)p eop %%Page: 9 9 9 8 bop 465 55 a Fl(F)o(ORMS)18 b(OF)e(HOPF)h(ALGEBRAS)g(AND)g(GALOIS)f (THEOR)m(Y)218 b(9)228 154 y Fu(map.\))27 b(This)18 b(is)h(ev)o(en)e (an)i(isomorphism)d(of)j Ft(A)p Fu(-algebras.)28 b(So)19 b(w)o(e)f(get)h(for)f(an)o(y)228 212 y Ft(A)p Fu(-algebra)e Ft(B)406 309 y(G)p Fr(j)458 316 y Fi(A)487 309 y Fu(\()p Ft(B)s Fu(\))578 295 y Fr(\030)578 311 y Fu(=)630 309 y Ft(k)r Fu(-)q Fr(A)p Fq(lg)p Fu(\()p Ft(H)811 288 y Fp(\003)831 309 y Ft(;)8 b(B)s Fu(\))926 295 y Fr(\030)926 311 y Fu(=)978 309 y Ft(A)p Fu(-)p Fr(A)p Fq(lg)q Fu(\()p Ft(A)i Fr(\012)h Ft(H)1266 288 y Fp(\003)1286 309 y Ft(;)d(B)s Fu(\))572 368 y Fr(\030)573 383 y Fu(=)625 381 y Ft(A)p Fu(-)p Fr(A)p Fq(lg)p Fu(\()p Ft(A)j Fr(\012)g Ft(A;)d(B)s Fu(\))999 368 y Fr(\030)1000 383 y Fu(=)1052 381 y Ft(k)r Fu(-)p Fr(A)p Fq(lg)q Fu(\()p Ft(A;)g(B)s Fu(\))1320 368 y Fr(\030)1320 383 y Fu(=)1372 381 y Ft(X)t Fr(j)1430 388 y Fi(A)1459 381 y Fu(\()p Ft(B)s Fu(\))p Ft(:)228 478 y Fu(It)17 b(is)h(no)o(w)g(easy)g(to)g(v)o(erify)e(that)i(this)g (is)g(an)g(isomorphism)d(of)k Ft(G)p Fr(j)1463 485 y Fi(A)1491 478 y Fu(-set)f(v)m(alued)228 536 y(functors.)1269 b Ff(\003)278 641 y Fu(The)18 b(translation)g(of)h(the)f(notion)g(of)h (principal)e(homogeneous)h(spaces)h(in)o(to)228 699 y(terms)j(of)h (Hopf)h(algebras)g(has)g(a)g(most)e(in)o(teresting)h(v)m(ariation.)43 b(Let)23 b Ft(A)g Fu(b)q(e)228 757 y(an)e Ft(H)344 739 y Fp(\003)364 757 y Fu(-como)q(dule)f(algebra.)36 b(Assume)19 b(no)o(w)j(that)f Ft(H)1253 739 y Fp(\003)1294 757 y Fu(is)g(\014nitely)e(generated)228 815 y(and)h(pro)s(jectiv)o(e)e(as)i (a)g Ft(k)r Fu(-mo)q(dule)f(and)h(that)g Ft(A)f Fu(is)g(faithfully)g (\015at.)31 b(The)20 b(dual)228 873 y Ft(H)28 b Fu(:=)c(Hom)473 880 y Fi(k)495 873 y Fu(\()p Ft(H)558 855 y Fp(\003)578 873 y Ft(;)8 b(k)r Fu(\))22 b(is)g(a)g(\014nitely)f(generated)h(pro)s (jectiv)o(e)f(co)q(comm)o(utativ)o(e)228 932 y(Hopf)h(algebra)i(whic)o (h)e(acts)h(on)g Ft(A)f Fu(b)o(y)g Ft(h)16 b Fr(\001)f Ft(t)24 b Fu(=)1169 894 y Fh(P)1230 932 y Ft(t)1248 939 y Fo(\()p Fi(A)p Fo(\))1303 932 y Ft(h)p Fu(\()p Ft(t)1368 939 y Fo(\()p Fi(H)1414 930 y Fg(\003)1431 939 y Fo(\))1447 932 y Fu(\).)41 b(Then)22 b(the)228 990 y(follo)o(wing)16 b(holds:)228 1067 y Fn(Theorem)f(and)j(De\014nition)e(2.5.)j Fq(Under)e(the)g(ab)n(ove)g(assumptions)g(the)f(fol-)228 1125 y(lowing)j(ar)n(e)e(e)n(quivalent:)321 1202 y Fu(a\))k Ft(A)j Fq(is)g(a)h(Hopf)f(Galois)h(extension)h(of)e Ft(k)j Fq(with)e(Hopf)f(algebr)n(a)h Ft(H)k Fq(\(or)385 1260 y(simply)17 b Ft(H)t Fq(-Galois\).)318 1319 y Fu(b\))k Ft( )15 b Fu(:)f Ft(A)s Fr(\012)s Ft(A)f Fr(3)h Ft(s)s Fr(\012)s Ft(t)g Fr(7!)803 1281 y Fh(P)863 1319 y Ft(st)904 1326 y Fo(\()p Fi(A)p Fo(\))963 1319 y Fr(\012)s Ft(t)1023 1326 y Fo(\()p Fi(H)1069 1317 y Fg(\003)1086 1326 y Fo(\))1116 1319 y Fr(2)g Ft(A)s Fr(\012)s Ft(H)1289 1301 y Fp(\003)1323 1319 y Fq(is)f(an)i(isomorphism.)324 1377 y Fu(c\))20 b Fq(Ther)n(e)14 b(is)h(a)f(faithful)r(ly)i(\015at)e(extension)j Ft(K)h Fq(of)d Ft(k)h Fq(with)f Ft(K)8 b Fr(\012)d Ft(A)1513 1363 y Fr(\030)1513 1379 y Fu(=)1565 1377 y Ft(K)j Fr(\012)d Ft(H)1702 1359 y Fp(\003)385 1435 y Fq(as)17 b Ft(K)e Fr(\012)c Ft(H)597 1417 y Fp(\003)617 1435 y Fq(-c)n(omo)n(dule)18 b(algebr)n(as.)318 1493 y Fu(d\))j Ft(\016)29 b Fu(:)d Ft(H)21 b Fr(\012)16 b Ft(A)26 b Fr(3)i Ft(h)16 b Fr(\012)g Ft(s)27 b Fr(7!)g Fu(\()p Ft(t)g Fr(7!)1082 1456 y Fh(P)1143 1493 y Ft(s)p Fu(\()p Ft(h)17 b Fr(\001)f Ft(t)p Fu(\)\))26 b Fr(2)i Fu(End)1490 1500 y Fi(k)1511 1493 y Fu(\()p Ft(A)p Fu(\))c Fq(is)h(an)385 1551 y(isomorphism)16 b(and)h Ft(A)g Fq(is)g(\014nitely)i(gener)n(ate)n(d)f(faithful)g(pr)n(oje)n (ctive)f(as)g(a)385 1609 y Ft(k)r Fq(-mo)n(dule.)324 1667 y Fu(e\))j Ft(k)g Fq(is)d(the)h(\014xring)577 1765 y Ft(A)614 1745 y Fi(H)661 1765 y Fu(:=)13 b Fr(f)p Ft(s)h Fr(2)g Ft(A)p Fr(j8)p Ft(h)e Fr(2)i Ft(H)k Fu(:)13 b Ft(h)e Fr(\001)g Ft(s)j Fu(=)g Ft(\017)p Fu(\()p Ft(h)p Fu(\))p Ft(s)p Fr(g)385 1862 y Fq(of)22 b Ft(A)f Fq(under)h(the)g (action)g(of)g Ft(H)j Fq(and)d(the)g(rings)g Ft(A)1360 1844 y Fi(H)1415 1862 y Fq(and)g Ft(A)p Fu(#)p Ft(H)j Fq(ar)n(e)385 1920 y(Morita)17 b(e)n(quivalent.)228 2025 y(Pr)n(o)n(of.)i Fu(a\):)g(A)11 b(Hopf)g(Galois)g(extension)g(is)g (de\014ned)g(to)h(b)q(e)f(one)g(of)h(the)f(equiv)m(alen)o(t)228 2083 y(conditions)22 b(b\))h(-)g(e\).)39 b(b\))23 b(implies)d(c\))i (with)g Ft(K)29 b Fu(=)24 b Ft(A)p Fu(.)40 b(The)22 b(equiv)m(alence)f (of)228 2141 y(b\))g(and)g(c\))f(is)h(the)g(preceding)f(Prop)q (osition.)36 b(The)20 b(equiv)m(alence)f(b)q(et)o(w)o(een)h(b\))228 2199 y(and)f(d\))f(is)g(a)h(simple)d(calculation)i(with)g(dual)g(bases) h(for)g Ft(H)j Fu(and)d Ft(H)1519 2181 y Fp(\003)1557 2199 y Fu(and)g(use)228 2258 y(of)h(faithful)g(\015atness.)34 b(e\))20 b(is)g(essen)o(tially)f(a)h(translation)h(of)g(d\))f(in)o(to)g (terms)f(of)228 2316 y(Morita)d(equiv)m(alences.)k(Detailed)15 b(pro)q(ofs)j(of)e(this)g(can)h(b)q(e)f(found)h(in)f([P].)82 b Ff(\003)278 2421 y Fu(There)11 b(are)h(v)m(arious)h(di\013eren)o(t)e (generalizations)h(of)g(Galois)g(extensions.)20 b(Non-)228 2479 y(comm)o(utativ)n(e)e(algebras)k(with)e(Hopf)h(algebras)h(acting)f (on)h(them)d(ha)o(v)o(e)h(b)q(een)228 2537 y(in)o(v)o(estigated.)f (Comm)o(utativ)o(e)13 b(algebras)j(with)f(\014nite)g(groups)i(acting)f (on)g(them)p eop %%Page: 10 10 10 9 bop 228 55 a Fl(10)545 b(BODO)13 b(P)m(AREIGIS)228 154 y Fu(ha)o(v)o(e)k(b)q(een)h(studied)g(in)g([CHR].)e(The)i (de\014nition)g(used)g(here)g(has)g(b)q(een)g(in)o(tro-)228 212 y(duced)e(in)f([CS])h(and)g(is)g(also)h(describ)q(ed)e(in)h([S1].) 21 b(Sp)q(ecial)15 b(instances)h(of)h(Galois)228 270 y(extensions)f(are)g(encluded)f(in)h(this)g(general)g(concept.)278 329 y(Let)j Ft(k)j Fu(b)q(e)e(a)g(\014eld)f(and)i Ft(H)j Fu(=)19 b Ft(k)r(G)h Fu(the)g(group)g(\(Hopf)s(\))h(algebra)f(of)g(a)g (\014nite)228 387 y(group.)36 b(Let)21 b Ft(K)k Fu(b)q(e)20 b(a)i(\014eld)e(extension)g(of)h Ft(k)i Fu(whic)o(h)d(is)h Ft(H)t Fu(-Galois.)36 b(Then)21 b Ft(G)228 445 y Fu(acts)c(b)o(y)f (automorphisms)g(on)h Ft(K)t Fu(.)23 b(F)l(urthermore)15 b(w)o(e)h(ha)o(v)o(e)g Ft(k)h Fu(=)d Fr(f)p Ft(s)h Fr(2)g Ft(K)t Fr(j8)p Ft(g)g Fr(2)228 503 y Ft(G)6 b Fu(:)16 b Ft(g)r Fu(\()p Ft(s)p Fu(\))e(=)g Ft(s)p Fr(g)f Fu(=)h Ft(K)612 485 y Fi(G)642 503 y Fu(.)21 b(Since)15 b([)p Ft(K)j Fu(:)13 b Ft(k)r Fu(])h(=)f Fr(j)p Ft(G)p Fr(j)k Fu(w)o(e)e(get)i(that)f Ft(K)21 b Fu(is)16 b(a)g("classical")228 561 y(Galois)g(extension)f(of)i Ft(k)h Fu(with)d(Galois)i(group)f Ft(G)p Fu(.)22 b(Con)o(v)o(ersely)14 b(if)i Ft(K)k Fu(is)15 b(a)i("clas-)228 619 y(sical")i(Galois)g(extension)g(of)g Ft(k)i Fu(with)e(Galois)g(group)h Ft(G)g Fu(then)f(b)o(y)f(Dedekind's) 228 677 y(lemm)o(a)13 b(and)j(d\))f(of)h(the)f(ab)q(o)o(v)o(e)g (Theorem)f Ft(K)19 b Fu(is)c(Hopf)g(Galois)h(with)f(Hopf)g(alge-)228 735 y(bra)i Ft(H)h Fu(=)13 b Ft(k)r(G)p Fu(.)278 793 y(Jacobson's)18 b(extension)e([J])g(of)h(Galois)g(theory)g(to)g(purely) f(inseparable)h(\014eld)228 852 y(extensions)c(can)h(b)q(e)g(incorp)q (orated)g(in)o(to)f(the)g(general)h(framew)o(ork)e(of)i(Hopf)f(Ga-)228 910 y(lois)j(theory)h(in)g(the)f(follo)o(wing)h(w)o(a)o(y)l(.)23 b(Jacobson)18 b(uses)f(restricted)f(Lie)h(algebras)228 968 y(acting)c(b)o(y)f(deriv)m(ations)h(on)g(purely)f(inseparable)h (\014eld)f(extensions)g(of)h(exp)q(onen)o(t)228 1026 y(one.)24 b(The)17 b(restricted)f(univ)o(ersal)g(en)o(v)o(eloping)g (algebras)i(of)g(the)e(restricted)g(Lie)228 1084 y(algebras)f(are)g (Hopf)f(algebras)h(and)h(the)e(action)g(extends)h(to)f(a)h(Hopf)g (Galois)g(ac-)228 1142 y(tion)h(on)i(the)e(same)g(extension.)22 b(Details)16 b(and)h(an)g(extension)f(to)h(a)g(larger)g(class)228 1200 y(of)d(purely)e(inseparable)h(\014eld)g(extensions)g(can)h(b)q(e)f (found)h(e.g.)20 b(in)13 b([S2])g(and)h([W].)278 1258 y(The)i(question)g(arises)g(whic)o(h)g(parts)h(of)g(the)f("classical")g (Galois)h(theory)f(can)228 1316 y(b)q(e)e(transferred)g(to)g(Hopf)g (Galois)h(theory)l(.)20 b(The)14 b(de\014nition)g(of)g(a)h(Hopf)f (subalge-)228 1375 y(bra)i Ft(H)358 1357 y Fp(0)384 1375 y Fr(\022)e Ft(H)20 b Fu(causes)c(some)g(problems)e(on)j(the)f (coalgebra)g(side.)21 b(If)16 b(w)o(e)f(alw)o(a)o(ys)228 1433 y(assume,)20 b(ho)o(w)o(ev)o(er,)g(that)h Ft(H)775 1415 y Fp(0)808 1433 y Fu(is)f(a)h(direct)f(summand)f(of)h Ft(H)25 b Fu(as)d(a)f Ft(k)r Fu(-mo)q(dule,)228 1491 y(these)e(problems)g(can)g(b)q(e)h(resolv)o(ed.)31 b(The)19 b(fundamen)o(tal)g(theorem)f(of)i(Galois)228 1549 y(theory)c(can)g(b)q (e)h(extended)e(to)228 1635 y Fn(Theorem)j(2.6.)j Fu([CS])d Fq(L)n(et)h Ft(K)k Fq(b)n(e)c(Hopf)g(Galois)g(with)g(Hopf)g(algebr)n(a) h Ft(H)t Fq(.)27 b(F)l(or)228 1693 y Ft(H)272 1675 y Fp(0)301 1693 y Fq(a)17 b(Hopf)h(sub)n(algebr)n(a)g(of)f Ft(H)22 b Fq(let)505 1808 y Fu(Fix)o(\()p Ft(H)639 1787 y Fp(0)651 1808 y Fu(\))14 b(:=)f Fr(f)p Ft(x)h Fr(2)g Ft(K)t Fr(j8)p Ft(h)e Fr(2)i Ft(H)1082 1787 y Fp(0)1108 1808 y Fu(:)g Ft(h)d Fr(\001)g Ft(x)i Fu(=)h Ft(\017)p Fu(\()p Ft(h)p Fu(\))p Ft(x)p Fr(g)p Ft(:)228 1922 y Fq(Then)260 2036 y Fu(Fix)t(:)i Fr(f)p Ft(H)435 2016 y Fp(0)461 2036 y Fr(\022)e Ft(H)t Fr(j)p Ft(H)616 2016 y Fp(0)645 2036 y Fq(Hopf)k(sub)n(algebr)n(a)g Fr(g)13 b(\000)-29 b(!)14 b(f)p Ft(L)p Fr(j)p Ft(k)i Fr(\022)d Ft(L)h Fr(\022)g Ft(K)21 b Fq(sub)n(algebr)n(a)d Fr(g)228 2150 y Fq(is)f(inje)n(ctive)i(and)f(inclusion-r)n(eversing.)278 2237 y Fu(W)l(e)g(sa)o(y)l(,)h(that)h(the)e(fundamen)o(tal)g(theorem)f (of)j(Galois)f(theory)g(hold)g(in)f(its)228 2295 y Fq(str)n(ong)24 b(form)p Fu(,)f(if)g(the)g(map)g(Fix)f(is)h(bijectiv)o(e.)40 b(This,)24 b(ho)o(w)o(ev)o(er,)g(is)f(not)g(the)228 2353 y(case)17 b(in)g(general,)f(as)i(w)o(e)f(will)f(see)h(b)q(elo)o(w.)23 b(There)17 b(is)g(another)h(deviation)e(from)228 2411 y(the)22 b("classical")h(Galois)g(theory)l(.)41 b(The)23 b(Hopf)f(algebra)i(acting)e(on)i(a)f(Galois)228 2469 y(extension)f Ft(K)27 b Fu(of)c Ft(k)i Fu(is)d(not)h(uniquely)f (determined.)38 b(Examples)21 b(ha)o(v)o(e)h(b)q(een)228 2527 y(kno)o(wn)16 b(for)h(inseparable)f(\014eld)f(extensions.)p eop %%Page: 11 11 11 10 bop 465 55 a Fl(F)o(ORMS)18 b(OF)e(HOPF)h(ALGEBRAS)g(AND)g (GALOIS)f(THEOR)m(Y)199 b(11)581 154 y Fu(3.)43 b Fs(Sep)m(arable)17 b(Field)i(Extensions)278 241 y Fu(W)l(e)12 b(giv)o(e)g(an)i(example)d (of)i(a)g(separable)g(\014eld)g(extension)f(whic)o(h)h(is)f(not)i (Galois)228 299 y(in)g(the)h(classical)f(sense,)g(but)h(whic)o(h)f(is)g (Hopf)h(Galois)g(with)g(t)o(w)o(o)f(di\013eren)o(t)g(Hopf)228 359 y(algebras.)21 b(Let)15 b Ft(K)j Fu(=)c Fn(Q)p Fu(\()708 335 y Fe(4)695 317 y Fr(p)p 737 317 25 2 v 42 x Fu(2\))h(and)g Ft(k)h Fu(=)e Fn(Q)p Fu(.)20 b(It)15 b(is)f(w)o(ell)g(kno)o(wn)h(that)g (this)f(is)h(not)228 417 y(a)h("classical")h(Galois)f(extension.)21 b(Let)674 503 y Ft(H)d Fu(=)c Fn(Q)p Fu([)p Ft(c;)8 b(s)p Fu(])p Ft(=)p Fu(\()p Ft(c)984 483 y Fo(2)1014 503 y Fu(+)j Ft(s)1086 483 y Fo(2)1117 503 y Fr(\000)g Fu(1)p Ft(;)d(cs)p Fu(\))228 594 y(with)18 b(the)h(coalgebra)g(structure)f(as) i(giv)o(en)d(in)i(part)g(I.)e(Abbreviate)h Ft(!)i Fu(:=)1655 571 y Fe(4)1643 553 y Fr(p)p 1684 553 V 1684 594 a Fu(2)q(.)228 652 y(Then)c(the)g(op)q(eration)h(of)g Ft(H)j Fu(on)d Ft(K)j Fu(is)c(giv)o(en)g(b)o(y)p 770 724 2 59 v 820 706 a(1)121 b Ft(!)106 b(!)1133 688 y Fo(2)1231 706 y Ft(!)1263 688 y Fo(3)p 683 726 600 2 v 770 734 2 9 v 683 775 a Fu(c)p 770 792 2 59 v 115 w(1)112 b(0)82 b Fr(\000)p Ft(!)1133 756 y Fo(2)1242 775 y Fu(0)685 833 y(s)p 770 850 V 116 w(0)g Fr(\000)p Ft(!)117 b Fu(0)82 b Ft(!)1250 815 y Fo(3)1270 833 y Ft(:)278 899 y Fu(If)15 b Ft(K)j Fu(=)13 b Ft(k)r Fu(\()496 876 y Fe(4)483 858 y Fr(p)p 525 858 25 2 v 41 x Fu(2)q(\))i(w)o(ere)g(a)h(classical)f (Galois)h(extension)f(for)g(example)f(o)o(v)o(er)g(the)228 958 y(base)k(\014eld)e Fn(Q)p Fu(\()p Ft(i)p Fu(\))i(then)f(the)g (Galois)h(group)g(is)f(cyclic)f(with)h(generator)h Ft(e)p Fu(.)24 b(Here)228 1016 y(the)16 b(generator)i Ft(e)e Fu(has)h(b)q(een)g(replaced)f(b)o(y)g(the)g(t)o(w)o(o)h(op)q(erators)h Ft(c)f Fu(and)g Ft(s)g Fu(whic)o(h)228 1074 y(op)q(erate)g Ft(k)r Fu(-linearly)e(and)h(according)h(to)g(the)f(rules)278 1161 y Ft(c)p Fu(\()p Ft(xy)r Fu(\))d(=)h Ft(c)p Fu(\()p Ft(x)p Fu(\))p Ft(c)p Fu(\()p Ft(y)r Fu(\))c Fr(\000)h Ft(s)p Fu(\()p Ft(x)p Fu(\))p Ft(s)p Fu(\()p Ft(y)r Fu(\))64 b(and)i Ft(s)p Fu(\()p Ft(xy)r Fu(\))13 b(=)h Ft(s)p Fu(\()p Ft(x)p Fu(\))p Ft(c)p Fu(\()p Ft(y)r Fu(\))c(+)h Ft(c)p Fu(\()p Ft(x)p Fu(\))p Ft(s)p Fu(\()p Ft(y)r Fu(\))p Ft(:)228 1247 y Fu(Similaritie)o(s)16 b(with)i(the)g(trigonometric)f (equalities)g(are)h(in)o(tended.)27 b(If)18 b(one)h(ex-)228 1306 y(tends)11 b(the)h(base)g(\014eld)f(from)f Fn(Q)i Fu(to)g Fn(Q)p Fu(\()p Ft(i)p Fu(\))f(then)g(\()p Fn(Q)p Fu(\()p Ft(i)p Fu(\))q Fr(\012)q Fn(Q)p Fu(\()1304 1283 y Fe(4)1290 1265 y Fr(p)p 1333 1265 V 1333 1306 a Fu(2\)\))j(:)f Fn(Q)p Fu(\()p Ft(i)p Fu(\))e(b)q(ecomes)228 1365 y(a)k(classical)g (Galois)h(extension)e(and)i(the)f(Hopf)g(algebra)h Ft(H)k Fu(is)15 b(extended)f(to)h(the)228 1424 y(group)k(ring)g Fn(Q)p Ft(C)548 1431 y Fo(4)568 1424 y Fu(.)29 b(By)18 b(further)g(extending)g(the)h(base)g(\014eld)f(to)h Fn(Q)p Fu(\()p Ft(i;)1563 1400 y Fe(4)1551 1382 y Fr(p)p 1592 1382 V 1592 1424 a Fu(2)q(\))f(the)228 1485 y(ring)12 b(extension)g Fn(Q)p Fu(\()p Ft(i;)650 1462 y Fe(4)637 1444 y Fr(p)p 679 1444 V 41 x Fu(2\))t Fr(\012)t Fn(Q)p Fu(\()843 1462 y Fe(4)831 1444 y Fr(p)p 872 1444 V 872 1485 a Fu(2\))h(b)q(ecomes)e(isomorphic)h(to)h(the)f(dual)h(of)f(the) 228 1547 y(extended)k(group)j(algebra)f(\()p Fn(Q)p Fu(\()p Ft(i;)881 1524 y Fe(4)868 1506 y Fr(p)p 910 1506 V 41 x Fu(2\))p Ft(C)988 1554 y Fo(4)1008 1547 y Fu(\))1027 1529 y Fp(\003)1046 1547 y Fu(.)26 b(This)17 b(isomorphism)f(is)h (compati-)228 1605 y(ble)h(with)h(the)g(como)q(dule)f(algebra)h (structure.)29 b(So)20 b(w)o(e)e(see)h(that)h(the)e(original)228 1663 y Ft(H)272 1645 y Fp(\003)292 1663 y Fu(-como)q(dule)f(algebra)i Ft(K)k Fu(is)18 b(a)h(form)e(of)i(the)f(trivial)f Ft(H)1311 1645 y Fp(\003)1331 1663 y Fu(-como)q(dule)h(algebra)228 1721 y Ft(H)272 1703 y Fp(\003)292 1721 y Fu(.)278 1780 y(One)c(can)g(sho)o(w)h(that)g(there)f(is)g(a)h(second)g(Hopf)f (algebra)h(o)o(v)o(er)f Fn(Q)g Fu(and)h(action)228 1839 y(on)k Ft(K)k Fu(=)18 b Fn(Q)p Fu(\()493 1815 y Fe(4)480 1797 y Fr(p)p 522 1797 V 42 x Fu(2)q(\))h(suc)o(h)f(that)i(the)f(setup) g(is)g(a)g(Hopf)g(Galois)g(extension.)29 b(The)228 1897 y(Hopf)16 b(algebra)h(is)662 1963 y Ft(H)h Fu(=)c Fn(Q)p Fu([)p Ft(c;)8 b(s)p Fu(])p Ft(=)p Fu(\()p Ft(s)974 1943 y Fo(2)1004 1963 y Fr(\000)j Fu(2)p Ft(c)1099 1943 y Fo(2)1130 1963 y Fu(+)g(2)p Ft(;)d(cs)p Fu(\))228 2040 y(with)16 b(the)g(action)p 766 2102 2 59 v 816 2085 a(1)84 b Ft(!)96 b(!)1082 2067 y Fo(2)1207 2085 y Ft(!)1239 2067 y Fo(3)p 679 2104 608 2 v 766 2112 2 9 v 679 2153 a Fu(c)p 766 2170 2 59 v 115 w(1)79 b(0)h Fr(\000)p Ft(!)1094 2135 y Fo(2)1212 2153 y Fu(0)681 2211 y(s)p 766 2228 V 116 w(0)66 b Ft(!)938 2193 y Fo(3)1056 2211 y Fu(0)98 b Fr(\000)p Fu(2)p Ft(!)r(:)278 2269 y Fu(The)13 b(maps)f Ft(c)h Fu(and)h Ft(s)f Fu(are)g Ft(k)r Fu(-linear)f(and)i(satisfy)f (the)g(m)o(ultiplic)o(ativ)o(e)c(relations)261 2377 y Ft(c)p Fu(\()p Ft(xy)r Fu(\))k(=)h Ft(c)p Fu(\()p Ft(x)p Fu(\))p Ft(c)p Fu(\()p Ft(y)r Fu(\))c Fr(\000)676 2344 y Fu(1)p 676 2366 25 2 v 676 2411 a(2)705 2377 y Ft(s)p Fu(\()p Ft(x)p Fu(\))p Ft(s)p Fu(\()p Ft(y)r Fu(\))65 b(and)g Ft(s)p Fu(\()p Ft(xy)r Fu(\))13 b(=)h Ft(c)p Fu(\()p Ft(x)p Fu(\))p Ft(s)p Fu(\()p Ft(y)r Fu(\))c(+)h Ft(s)p Fu(\()p Ft(x)p Fu(\))p Ft(c)p Fu(\()p Ft(y)r Fu(\))p Ft(:)228 2479 y Fu(T)l(o)19 b(see)g(that)g(this)g(giv)o(es)g(a)g(Hopf)g (Galois)g(extension)g(one)g(has)h(to)f(extend)f(the)228 2537 y(base)e(\014eld)g(to)h Fn(Q)p Fu(\()549 2498 y Fr(p)p 590 2498 64 2 v 590 2537 a(\000)p Fu(2)q(\))f(and)h(then)f(pro)q (cede)g(as)h(ab)q(o)o(v)o(e.)p eop %%Page: 12 12 12 11 bop 228 55 a Fl(12)545 b(BODO)13 b(P)m(AREIGIS)278 154 y Fu(This)k(is)h(an)g(example)e(of)i(a)g Ft(k)r Fu(-algebra)g(whic) o(h)f(is)h(a)g(Hopf)g(Galois)g(extension)228 212 y(with)h(t)o(w)o(o)g (di\013eren)o(t)f(Hopf)h(algebras.)31 b(W)l(e)19 b(will)f(see)h (further)g(do)o(wn)h(that)f(this)228 270 y(will)g(happ)q(en)j Fq(very)e Fu(often.)35 b(Ev)o(en)20 b(the)g("classical")h(Galois)g (extensions)f(often)228 329 y(ha)o(v)o(e)c(more)f(than)j(one)f(Hopf)g (algebra)g(for)g(whic)o(h)f(they)h(are)g(Hopf)g(Galois.)23 b(On)228 387 y(the)15 b(other)g(hand)h(there)f(are)g(separable)h (\014eld)f(extensions)g(whic)o(h)f(are)h(not)h(Hopf)228 445 y(Galois)f(at)f(all.)20 b(The)15 b(separable)f(\014eld)g (extensions)g(whic)o(h)g(are)g(Hopf)g(Galois)h(can)228 503 y(b)q(e)h(classi\014ed)g(b)o(y)g(the)g(follo)o(wing)g(theorem.)278 561 y(T)l(o)g(form)o(ulate)e(the)i(theorem)e(w)o(e)h(\014x)h(the)g (follo)o(wing)f(notation.)22 b(Let)16 b Ft(K)k Fu(b)q(e)c(a)228 619 y(\014nite)f(separable)i(\014eld)e(extension)h(of)h Ft(k)r Fu(.)k(Assume)481 705 y(~)468 718 y Ft(K)45 b Fu(=)14 b(normal)h(closure)h(of)h Ft(K)j Fu(o)o(v)o(er)c Ft(k)r(;)474 781 y(G)42 b Fu(=)14 b(Aut\()721 768 y(~)708 781 y Ft(K)t(=k)r Fu(\))p Ft(;)463 844 y(G)501 826 y Fp(0)554 844 y Fu(=)g(Aut\()721 831 y(~)708 844 y Ft(K)t(=K)t Fu(\))p Ft(;)480 902 y(S)44 b Fu(=)14 b Ft(G=G)706 884 y Fp(0)784 902 y Fu(\(left)i(cosets\),)473 960 y Ft(B)44 b Fu(=)14 b(P)o(erm)n(\()p Ft(S)s Fu(\))65 b(\(group)17 b(of)g(p)q(erm)o(utations)e(of)i Ft(S)s Fu(\))p Ft(:)228 1058 y Fn(Theorem)f(3.1.)j Fu([GP])e Fq(Under)g(the)g(assumptions)g (made)g(ab)n(ove)h(the)f(fol)r(lowing)228 1116 y(ar)n(e)g(e)n (quivalent)321 1194 y Fu(a\))k Fq(Ther)n(e)c(is)g(a)h(Hopf)f Ft(k)r Fq(-algebr)n(a)i Ft(H)i Fq(such)d(that)g Ft(K)q(=k)i Fq(is)d Ft(H)t Fq(-Galois.)318 1252 y Fu(b\))k Fq(Ther)n(e)16 b(is)h(a)f(r)n(e)n(gular)g(sub)n(gr)n(oup)g Ft(N)j Fr(\022)14 b Ft(B)19 b Fq(such)e(that)f(the)h(sub)n(gr)n(oup)f Ft(G)e Fr(\022)385 1310 y Ft(B)20 b Fq(normalizes)e Ft(N)5 b Fq(.)278 1388 y Fu(The)15 b(examples)f(giv)o(en)g(ab)q(o)o(v)o(e)i(are) g(of)f(a)h(rather)g(sp)q(ecial)f(t)o(yp)q(e)g(whic)o(h)g(w)o(e)g(call) 228 1446 y("almost)j(classical")g(Hopf)g(Galois)h(extensions.)28 b(They)18 b(are)h(c)o(haracterized)e(b)o(y)228 1504 y(the)f(follo)o (wing)228 1582 y Fn(Theorem)g(3.2.)k Fu([GP])d Fq(The)h(fol)r(lowing)i (c)n(onditions)d(ar)n(e)g(e)n(quivalent:)321 1659 y Fu(a\))k Fq(Ther)n(e)c(exists)g(a)g(Galois)g(extension)i Ft(E)s(=k)g Fq(such)e(that)g Ft(E)c Fr(\012)d Ft(K)21 b Fq(is)16 b(a)h(\014eld)385 1721 y(c)n(ontaining)636 1709 y Fu(~)623 1721 y Ft(K)t Fq(.)318 1781 y Fu(b\))k Fq(Ther)n(e)c(exists)h(a)g (Galois)f(extension)i Ft(E)s(=k)h Fq(such)e(that)g Ft(E)c Fr(\012)d Ft(K)18 b Fu(=)1610 1768 y(~)1597 1781 y Ft(K)5 b Fq(.)324 1839 y Fu(c\))20 b Ft(G)423 1821 y Fp(0)453 1839 y Fq(has)d(a)g(normal)h(c)n(omplement)g Ft(N)23 b Fq(in)18 b Ft(G)p Fq(.)318 1897 y Fu(d\))j Fq(Ther)n(e)g(exists)h(a)f (r)n(e)n(gular)f(sub)n(gr)n(oup)h Ft(N)26 b Fr(\022)20 b Ft(B)k Fq(normalize)n(d)d(by)g Ft(G)h Fu(and)385 1955 y(con)o(tained)17 b Fq(in)h Ft(G)p Fq(.)278 2033 y Fu(The)e(last)g (condition)f(of)i(this)e(theorem)g(sho)o(ws)i(that)f(w)o(e)f(are)h (indeed)g(talking)228 2091 y(ab)q(out)k(Hopf)f(Galois)h(extensions.)29 b(These)19 b(extensions)f(are)i(particularly)e(w)o(ell)228 2149 y(b)q(eha)o(v)o(ed)12 b(b)q(ecause)g(they)g(satisfy)g(the)g (fundamen)o(tal)f(theorem)g(of)i(Galois)f(theory)228 2207 y(in)k(its)g(strong)h(form.)228 2285 y Fn(Theorem)23 b(3.3.)g Fu([GP])g Fq(If)f Ft(K)q(=k)k Fq(is)d(almost)g(classic)n(al)r (ly)h(Galois,)g(then)g(ther)n(e)228 2343 y(is)d(a)h(Hopf)g(algebr)n(a)g Ft(H)k Fq(such)c(that)g Ft(K)q(=k)j Fq(is)c Ft(H)t Fq(-Galois)i(and)f (the)g(map)f Fu(Fix)g Fq(is)228 2401 y(bije)n(ctive.)278 2479 y Fu(The)13 b(am)o(biguit)o(y)d(of)j(the)g(Hopf)g(algebra)g (acting)g(on)g(a)h(Hopf)e(Galois)i(extension)228 2537 y(is)i(exp)q(osed)g(in)g(the)g(follo)o(wing)p eop %%Page: 13 13 13 12 bop 465 55 a Fl(F)o(ORMS)18 b(OF)e(HOPF)h(ALGEBRAS)g(AND)g (GALOIS)f(THEOR)m(Y)199 b(13)228 154 y Fn(Theorem)22 b(3.4.)g Fu([GP])g Fq(A)o(ny)g(classic)n(al)h(Galois)f(extension)i Ft(K)q(=k)g Fq(c)n(an)f(b)n(e)f(en-)228 212 y(dowe)n(d)g(with)h(an)f Ft(H)t Fq(-Galois)h(structur)n(e)f(such)h(that)f(the)h(fol)r(lowing)i (variant)d(of)228 270 y(the)c(fundamental)h(the)n(or)n(em)e(holds:)22 b(Ther)n(e)c(is)f(a)h(c)n(anonic)n(al)g(bije)n(ction)h(b)n(etwe)n(en) 228 329 y(Hopf)e(sub)n(algebr)n(as)h(of)f Ft(H)22 b Fq(and)c Fu(normal)e Fq(interme)n(diate)i(\014elds)h Ft(k)d Fr(\022)d Ft(E)k Fr(\022)d Ft(K)t Fq(.)278 401 y Fu(One)21 b(of)i(the)e(simplest) f(examples)g(of)j(a)f(classical)f(Galois)i(extension)e(with)228 468 y(this)16 b(new)h Ft(H)t Fu(-Galois)g(structure)g(is)f(the)g(follo) o(wing.)22 b(Let)17 b Ft(\020)k Fu(b)q(e)c(a)g(3)1465 449 y(rd)1530 468 y(primitiv)o(e)228 527 y(ro)q(ot)g(of)g(unit)o(y)f (and)h(let)e Ft(!)i Fu(:=)807 503 y Fe(3)795 485 y Fr(p)p 836 485 25 2 v 836 527 a Fu(2)q(.)k(Then)c Ft(K)h Fu(=)c Fn(Q)p Fu(\()p Ft(!)r(;)8 b(\020)t Fu(\))17 b(is)f(a)h(classical)f (Galois)228 585 y(extension)f(of)i Fn(Q)f Fu(with)g(Galois)g(group)i Ft(S)986 592 y Fo(3)1005 585 y Fu(.)j(It)16 b(is)g(also)h(Hopf)f (Galois)g(with)g(Hopf)228 643 y(algebra)266 730 y Ft(H)i Fu(=)c Fn(Q)p Fu([)p Ft(c;)8 b(s;)g(t)p Fu(])p Ft(=)p Fu(\()p Ft(c)p Fu(\()p Ft(c)h Fr(\000)i Fu(1\)\()p Ft(c)h Fu(+)f(1\))p Ft(;)d Fu(2)p Ft(c)969 709 y Fo(2)1000 730 y Fu(+)j Ft(st)g Fu(+)g Ft(ts)f Fr(\000)h Fu(2)p Ft(;)d(cs;)g(sc;)g (ct;)g(tc;)g(s)1574 709 y Fo(2)1593 730 y Ft(;)g(t)1633 709 y Fo(2)1652 730 y Fu(\))p Ft(:)228 817 y Fu(The)16 b(action)g(of)h Ft(H)j Fu(on)d Ft(K)k Fu(is)16 b(describ)q(ed)f(b)o(y)h (the)g(table)p 868 889 2 59 v 918 871 a(1)83 b Ft(!)k(\020)p 781 890 405 2 v 868 899 2 9 v 781 939 a Fu(c)p 868 957 2 59 v 115 w(1)79 b(0)g Ft(\020)1149 921 y Fo(2)783 997 y Fu(s)p 868 1015 V 116 w(0)65 b Ft(!)1039 979 y Fo(2)1135 997 y Fu(0)784 1056 y(t)p 868 1073 V 115 w(0)79 b(0)h(0.)278 1114 y(The)16 b(action)g(of)h(the)f(three)f(generating)i(elemen)o(ts)c Ft(c;)8 b(s;)g(t)16 b Fu(on)h Ft(K)j Fu(sati\014es)523 1197 y Ft(c)p Fu(\()p Ft(xy)r Fu(\))41 b(=)14 b Ft(c)p Fu(\()p Ft(x)p Fu(\))p Ft(c)p Fu(\()p Ft(y)r Fu(\))c(+)965 1178 y Fo(1)p 965 1186 18 2 v 965 1214 a(2)988 1197 y Ft(s)p Fu(\()p Ft(x)p Fu(\))p Ft(t)p Fu(\()p Ft(y)r Fu(\))g(+)1223 1178 y Fo(1)p 1223 1186 V 1223 1214 a(2)1245 1197 y Ft(t)p Fu(\()p Ft(x)p Fu(\))p Ft(s)p Fu(\()p Ft(y)r Fu(\))p Ft(;)521 1256 y(s)p Fu(\()p Ft(xy)r Fu(\))41 b(=)14 b Ft(c)p Fu(\()p Ft(x)p Fu(\))p Ft(s)p Fu(\()p Ft(y)r Fu(\))c(+)h Ft(s)p Fu(\()p Ft(x)p Fu(\))p Ft(c)p Fu(\()p Ft(y)r Fu(\))f(+)1200 1236 y Fo(1)p 1200 1244 V 1200 1273 a(2)1223 1256 y Ft(t)p Fu(\()p Ft(x)p Fu(\))p Ft(t)p Fu(\()p Ft(y)r Fu(\))p Ft(;)527 1314 y(t)p Fu(\()p Ft(xy)r Fu(\))40 b(=)14 b Ft(c)p Fu(\()p Ft(x)p Fu(\))p Ft(t)p Fu(\()p Ft(y)r Fu(\))c(+)h Ft(t)p Fu(\()p Ft(x)p Fu(\))p Ft(c)p Fu(\()p Ft(y)r Fu(\))f(+)h Ft(s)p Fu(\()p Ft(x)p Fu(\))p Ft(s)p Fu(\()p Ft(y)r Fu(\))p Ft(:)278 1399 y Fu(W)l(e)g(\014nish)h(this)g(paragraph)h(on)g (separable)f(\014eld)f(extensions)g(whic)o(h)g(are)h(Hopf)228 1457 y(Galois)f(b)o(y)g(giving)g(a)g(family)e(of)i(examples)e(of)j (separable)f(\014eld)f(extensions)h(whic)o(h)228 1515 y(are)22 b(not)h(Hopf)f(Galois:)33 b(no)23 b(\014eld)e(extension)h Ft(K)k Fu(o)o(v)o(er)21 b Fn(Q)i Fu(of)f(degree)g(5)g(with)228 1575 y(automorphism)15 b(group)i Ft(S)719 1582 y Fo(5)755 1575 y Fu(of)823 1562 y(~)810 1575 y Ft(K)t(=k)i Fu(can)d(b)q(e)h(Hopf) f(Galois)h([GP].)543 1751 y(4.)43 b Fs(Hopf)17 b(Algebra)g(F)o(orms)i (revisited)278 1838 y Fu(Man)o(y)13 b(of)i(the)e(follo)o(wing)h (results)g(ha)o(v)o(e)f(b)q(een)h(obtained)h(in)e(co)q(op)q(eration)j (and)228 1896 y(discussions)h(with)f(studen)o(ts)h(and)g(collegues)f (of)h(mine.)k(In)16 b(particular)g(I)h(grate-)228 1954 y(fully)k(ac)o(kno)o(wledge)h(the)g(co)q(op)q(eration)h(of)g(C.)f (Greither,)h(R.)f(Haggenm)q(\177)-26 b(uller,)228 2012 y(and)17 b(C.)f(W)l(enninger.)278 2070 y(The)f(tec)o(hniques)g(to)h (pro)o(v)o(e)f(Theorem)f(1.2)i(can)g(b)q(e)g(used)g(to)g(calculate)f (more)228 2129 y(forms)h(of)h(group)h(rings.)24 b(The)17 b(adv)m(an)o(tage)h(in)f(the)g(pro)q(of)h(of)g(Theorem)d(1.2)j(w)o(as) 228 2187 y(that)h(all)g(quadratic)g(extensions)g(of)g(a)h(comm)o (utativ)n(e)c(ring)j(can)h(b)q(e)f(explicitly)228 2245 y(describ)q(ed)g(if)g(the)h(ring)g(satis\014es)g(only)g(minor)e (conditions)i([Sm].)30 b(If)19 b(2)h(is)g(not)228 2303 y(a)g(zero)f(divisor)h(in)f Ft(k)j Fu(and)e(if)f(Pic)863 2311 y Fo(\(2\))910 2303 y Fu(\()p Ft(k)r Fu(\))g(=)h(0)g(then)g(all)f (quadratic)h(extensions)228 2363 y(of)g Ft(k)i Fu(are)d(free)g(and)h (can)g(b)q(e)g(describ)q(ed)f(as)i Ft(K)i Fu(=)d Ft(k)r Fu([)p Ft(x)p Fu(])p Ft(=)p Fu(\()p Ft(x)1335 2344 y Fo(2)1367 2363 y Fr(\000)13 b Ft(ax)g Fr(\000)g Ft(b)p Fu(\))20 b(where)228 2421 y Ft(a)254 2403 y Fo(2)275 2421 y Fu(+)r(4)p Ft(b)14 b Fu(=)g Ft(u)d Fu(is)h(a)g(unit)g(in)f Ft(k)r Fu(.)20 b(The)12 b(non-trivial)f(automorphism)f(is)i Ft(f)5 b Fu(\()p Ft(x)p Fu(\))14 b(=)g Ft(a)r Fr(\000)r Ft(x)p Fu(.)228 2479 y(This)f(information)f(w)o(as)i(translated)f(in)o (to)g(terms)f(of)h(Hopf)g(algebra)h(forms)e(using)228 2537 y(the)k(follo)o(wing)p eop %%Page: 14 14 14 13 bop 228 55 a Fl(14)545 b(BODO)13 b(P)m(AREIGIS)228 154 y Fn(Theorem)24 b(4.1.)f Fu([HP])g Fq(L)n(et)g Ft(G)h Fq(b)n(e)g(a)g(\014nitely)h(gener)n(ate)n(d)f(gr)n(oup)f(with)h (\014nite)228 212 y(automorphism)13 b(gr)n(oup)g Ft(F)20 b Fu(=)14 b Fr(G)m Fq(rp)o(-)q Fu(Aut\()p Ft(G)p Fu(\))g Fq(Then)g(ther)n(e)g(is)g(a)g(bije)n(ction)h(b)n(etwe)n(en)228 270 y Fu(Gal\()p Ft(k)r(;)8 b(F)f Fu(\))p Fq(,)20 b(the)g(set)g(of)g (ismorphism)f(classes)i(of)e Ft(F)7 b Fq(-Galois)20 b(extensions)h(of)f Ft(k)r Fq(,)228 329 y(and)j Fu(Hopf\()p Ft(k)r(G)p Fu(\))p Fq(,)i(the)f(set)f(of)g(Hopf)g(algebr)n(a)h(forms)e(of)h Ft(k)r(G)p Fq(.)40 b(This)23 b(bije)n(ction)228 387 y(asso)n(ciates)17 b(with)h(e)n(ach)g Ft(F)7 b Fq(-Galois)17 b(extension)i Ft(K)j Fq(of)17 b Ft(k)i Fq(the)f(Hopf)g(algebr)n(a)362 482 y Ft(H)g Fu(=)472 427 y Fh(n)505 435 y(X)586 482 y Ft(c)607 489 y Fi(g)627 482 y Ft(g)e Fr(2)e Ft(K)t(G)p 822 494 4 59 v 43 w Fr(8)p Ft(f)k Fr(2)c Ft(F)e Fu(:)1039 435 y Fh(X)1120 482 y Ft(f)5 b Fu(\()p Ft(c)1189 489 y Fi(g)1209 482 y Fu(\))p Ft(f)g Fu(\()p Ft(g)r Fu(\))14 b(=)1386 435 y Fh(X)1466 482 y Ft(c)1487 489 y Fi(g)1507 482 y Ft(g)1532 427 y Fh(o)1574 482 y Ft(:)228 577 y Fq(F)l(urthermor)n(e)i Ft(H)22 b Fq(is)17 b(a)h Ft(K)t Fq(-form)f(of)g Ft(k)r(G)h Fq(by)g(the)g(isomorphism)596 659 y Ft(!)7 b Fu(:)17 b Ft(H)f Fr(\012)10 b Ft(K)828 646 y Fr(\030)829 662 y Fu(=)881 659 y Ft(K)t(G;)58 b(!)r Fu(\()p Ft(h)11 b Fr(\012)g Ft(a)p Fu(\))j(=)f Ft(ah:)278 742 y Fu(On)e(the)g(other)g(hand)h(it)e(is)h(not)h(trivial)e(to)h (describ)q(e)g Ft(F)c Fu(-Galois)k(extensions)f(of)i(a)228 800 y(\014eld)h Ft(k)r Fu(.)21 b(They)13 b(are)h(not)g(just)h(the)e (classical)g(Galois)i(\014eld)e(extensions)h(of)g Ft(k)r Fu(.)20 b(The)228 858 y(simple)11 b(example)h(of)i(the)f(trivial)f Ft(F)7 b Fu(-Galois)14 b(extension)f Ft(k)1284 840 y Fi(F)1327 844 y Fr(\030)1328 860 y Fu(=)1380 858 y Ft(k)7 b Fr(\002)f Ft(:)i(:)g(:)d Fr(\002)h Ft(k)15 b Fu(is)f(not)228 916 y(a)19 b(\014eld.)28 b(Actually)17 b Ft(F)7 b Fu(-Galois)19 b(extensions)f(are)h(just)g(Hopf)f(Galois)i(extensions)228 974 y(with)d(Hopf)h(algebra)g Ft(k)r(F)24 b Fu([CHR,)17 b(Thm)g(1.3].)25 b(Arbitrary)17 b(comm)o(utativ)o(e)d(rings)228 1032 y Ft(K)j Fu(are)d(admitted)e(as)i(Galois)g(extensions.)19 b(The)14 b(action)f(of)h(the)f(group)i Ft(F)20 b Fu(on)13 b(the)228 1090 y(extension)g Ft(K)18 b Fu(b)o(y)c(di\013eren)o(t)f (elemen)o(ts)f Ft(f)s(;)c(f)1032 1072 y Fp(0)1057 1090 y Fu(has)15 b(to)f(b)q(e)h("strongly)f(distict",)g(i.e.)228 1149 y(for)e(ev)o(ery)e(idemp)q(oten)o(t)g Ft(e)k Fr(2)g Ft(K)i Fu(there)11 b(is)g(an)i Ft(x)g Fr(2)h Ft(K)i Fu(suc)o(h)c(that)g Ft(f)5 b Fu(\()p Ft(x)p Fu(\))p Ft(e)14 b Fr(6)p Fu(=)f Ft(f)1608 1130 y Fp(0)1620 1149 y Fu(\()p Ft(x)p Fu(\))p Ft(e)p Fu(.)228 1207 y(This)j(is)g(the)g(k)o(ey)f(to)i(the)f(follo)o (wing)228 1277 y Fn(Theorem)k(4.2.)i Fq(L)n(et)e Ft(F)27 b Fq(b)n(e)21 b(a)f(\014nite)i(gr)n(oup)e(and)h Ft(k)i Fq(a)d(\014eld.)33 b Ft(K)q(=k)23 b Fq(is)e(an)f Ft(F)7 b Fq(-)228 1335 y(Galois)17 b(extension)j(if)d(and)g(only)h(if)797 1486 y Ft(K)855 1472 y Fr(\030)856 1488 y Fu(=)959 1411 y Fi(n)8 b Fu(times)908 1439 y Fh(z)p 930 1439 78 6 v 78 w(}|)p 1052 1439 V 78 w({)908 1486 y Ft(L)j Fr(\002)g Ft(:)d(:)g(:)i Fr(\002)h Ft(L)228 1568 y Fq(wher)n(e)23 b Ft(L=k)j Fq(is)d(a)g Ft(U)5 b Fq(-Galois)24 b(\014eld)g(extension)h (with)f Ft(U)29 b Fr(\022)24 b Ft(F)30 b Fq(a)23 b(sub)n(gr)n(oup)f(of) 228 1626 y(index)c Ft(n)p Fq(.)228 1714 y(Pr)n(o)n(of.)h Fu(Let)14 b Ft(K)q(=k)j Fu(b)q(e)d(an)h Ft(F)7 b Fu(-Galois)14 b(extension.)20 b Ft(K)e Fu(is)c(a)h(comm)o(utativ)n(e)c(separa-)228 1772 y(ble)j Ft(k)r Fu(-algebra)i(b)o(y)e([CHR)h(Thm)f(1.3])g(hence)h (is)f(a)i(pro)q(duct)f Ft(K)1391 1758 y Fr(\030)1392 1774 y Fu(=)1444 1772 y Ft(L)1477 1779 y Fo(1)1505 1772 y Fr(\002)9 b Ft(:)f(:)g(:)f Fr(\002)i Ft(L)1699 1779 y Fi(n)228 1830 y Fu(of)18 b(separable)g(\014eld)f(extensions)h Ft(L)878 1837 y Fi(i)892 1830 y Ft(=k)r Fu(.)27 b(The)18 b(automorphisms)f(in)g Ft(F)24 b Fu(map)17 b(the)228 1888 y(primitiv)o(e)10 b(idemp)q(oten)o(ts)j(to)h(primitiv)o(e)c(idemp) q(oten)o(ts)j(and)i Ft(F)20 b Fu(op)q(erates)15 b(transi-)228 1946 y(tiv)o(ely)g(on)j(the)f(set)g(primitiv)o(e)d(idemp)q(oten)o(ts,)h (since)i(the)g(sum)f(of)i(idemp)q(oten)o(ts)228 2004 y(in)e(an)g(orbit)g(is)g(in)g(the)f(\014xed)h(\014eld.)21 b(F)l(or)16 b(an)o(y)g(t)o(w)o(o)g(idemp)q(oten)o(ts)e Ft(e)1473 2011 y Fi(i)1503 2004 y Fu(and)j Ft(e)1621 2011 y Fi(j)1654 2004 y Fu(the)228 2063 y(automorphism)f Ft(f)22 b Fu(of)c Ft(F)24 b Fu(mapping)17 b Ft(e)936 2070 y Fi(i)967 2063 y Fu(to)g Ft(e)1050 2070 y Fi(j)1086 2063 y Fu(also)h(maps)e Ft(L)1346 2070 y Fi(i)1378 2063 y Fu(to)i Ft(L)1472 2070 y Fi(j)1490 2063 y Fu(.)25 b(Hence)16 b Ft(L)1708 2070 y Fi(i)228 2121 y Fu(is)k(isomorphic)g(to)h(a)g (sub\014eld)f(of)i Ft(L)919 2128 y Fi(j)937 2121 y Fu(.)35 b(By)20 b(symmetry)d(all)j(the)h(\014elds)f Ft(L)1622 2128 y Fi(i)1657 2121 y Fu(are)228 2179 y(m)o(utually)14 b(isomorphic.)22 b(The)16 b(stabilizer)g Ft(U)k Fr(\022)14 b Ft(F)23 b Fu(of)18 b Ft(e)1253 2186 y Fo(1)1289 2179 y Fu(acts)f(as)g(Galois)g(group)228 2237 y(on)g Ft(L)329 2244 y Fo(1)348 2237 y Ft(=k)i Fu(since)d(it)f(acts)i(strongly)g (distinctly)d(and)j Fr(j)p Ft(U)5 b Fr(j)14 b Fu(=)g([)p Ft(L)f Fu(:)h Ft(k)r Fu(].)278 2295 y(Con)o(v)o(ersely)19 b(let)g Ft(U)26 b Fr(\022)20 b Ft(G)h Fu(b)q(e)g(a)f(subgroup)i(and)f Ft(L)g Fu(:)f Ft(k)i Fu(b)q(e)f Ft(U)5 b Fu(-Galois.)34 b(Let)228 2353 y Ft(g)251 2360 y Fo(1)271 2353 y Ft(;)8 b(:)g(:)g(:)f(;)h(g)403 2360 y Fi(n)446 2353 y Fu(b)q(e)20 b(a)f(set)g(of)h(represen)o(tativ)o(es)d(for)j Ft(G=U)25 b Fu(=)19 b Fr(f)p Ft(g)1335 2360 y Fo(1)1355 2353 y Ft(U;)8 b(:)g(:)g(:)f(;)h(g)1520 2360 y Fi(n)1544 2353 y Ft(U)d Fr(g)p Fu(.)30 b(Let)228 2411 y Ft(K)21 b Fu(=)d Ft(L)13 b Fr(\002)f Ft(:)c(:)g(:)k Fr(\002)g Ft(L)19 b Fu(with)f(idemp)q(otens)f Ft(e)1013 2418 y Fo(1)1033 2411 y Ft(;)8 b(:)g(:)g(:)f(;)h(e)1165 2418 y Fi(n)1188 2411 y Fu(.)28 b(De\014ne)18 b(the)g(action)h Ft(\033)7 b Fu(:)17 b Ft(G)228 2469 y Fr(\000)-30 b(!)19 b Ft(S)336 2476 y Fi(n)380 2469 y Fu(b)o(y)g Ft(\033)r Fu(\()p Ft(g)r Fu(\)\()p Ft(i)p Fu(\))g(=)h Ft(j)j Fu(if)c Ft(g)r(g)815 2476 y Fi(i)830 2469 y Ft(U)25 b Fu(=)20 b Ft(g)969 2476 y Fi(j)988 2469 y Ft(U)25 b Fu(the)19 b(regular)h(represen)o(tation)g (of)g Ft(G)228 2529 y Fu(on)d Ft(G=U)5 b Fu(.)25 b(W)l(e)16 b(de\014ne)h Ft(g)r Fu(\()p Ft(l)q(e)744 2536 y Fi(i)757 2529 y Fu(\))e(:=)g Ft(g)883 2509 y Fp(\000)p Fo(1)881 2545 y Fi(\033)q Fo(\()p Fi(g)q Fo(\)\()p Fi(i)p Fo(\))990 2529 y Ft(g)r(g)1038 2536 y Fi(i)1052 2529 y Fu(\()p Ft(l)q Fu(\))p Ft(e)1129 2537 y Fi(\033)q Fo(\()p Fi(g)q Fo(\)\()p Fi(i)p Fo(\))1237 2529 y Fu(.)23 b(Observ)o(e)16 b(that)h Ft(g)r(g)1616 2536 y Fi(i)1631 2529 y Ft(U)j Fu(=)p eop %%Page: 15 15 15 14 bop 465 55 a Fl(F)o(ORMS)18 b(OF)e(HOPF)h(ALGEBRAS)g(AND)g (GALOIS)f(THEOR)m(Y)199 b(15)228 155 y Ft(g)251 162 y Fi(\033)q Fo(\()p Fi(g)q Fo(\)\()p Fi(i)p Fo(\))p Fi(U)407 155 y Fu(implies)18 b Ft(u)605 162 y Fi(g)q(;i)668 155 y Fu(:=)j Ft(g)766 134 y Fp(\000)p Fo(1)764 170 y Fi(\033)q Fo(\()p Fi(g)q Fo(\)\()p Fi(i)p Fo(\))873 155 y Ft(g)r(g)921 162 y Fi(i)956 155 y Fr(2)h Ft(U)5 b Fu(.)34 b(Then)21 b(the)f(\014xring)h(of)g Ft(K)j Fu(under)228 224 y(the)17 b(action)h(of)f Ft(G)h Fu(is)f Ft(k)r Fu(,)h(for)f(let)829 186 y Fh(P)890 224 y Ft(l)905 231 y Fi(i)919 224 y Ft(e)942 231 y Fi(i)971 224 y Fr(2)f Ft(K)1065 206 y Fi(G)1095 224 y Fu(.)25 b(Then)17 b(for)h(all)f Ft(g)h Fr(2)e Ft(G)i Fu(w)o(e)f(ha)o(v)o(e)228 245 y Fh(P)289 282 y Ft(u)317 289 y Fi(g)q(;i)358 282 y Fu(\()p Ft(l)392 289 y Fi(i)406 282 y Fu(\))p Ft(e)448 290 y Fi(\033)q Fo(\()p Fi(g)q Fo(\)\()p Fi(i)p Fo(\))574 282 y Fu(=)631 245 y Fh(P)691 282 y Ft(l)706 289 y Fi(i)720 282 y Ft(e)743 289 y Fi(i)757 282 y Fu(.)29 b(F)l(or)19 b Ft(g)i Fu(:=)c Ft(g)1026 289 y Fi(i)1041 282 y Ft(ug)1094 262 y Fp(\000)p Fo(1)1092 295 y Fi(i)1160 282 y Fu(w)o(e)h(get)h Ft(g)r(g)1366 289 y Fi(i)1381 282 y Ft(U)24 b Fu(=)18 b Ft(g)1517 289 y Fi(i)1531 282 y Ft(U)5 b Fu(,)20 b(hence)228 344 y Ft(\033)r Fu(\()p Ft(g)r Fu(\)\()p Ft(i)p Fu(\))13 b(=)h Ft(i)h Fu(and)h Ft(u)595 351 y Fi(g)q(;i)651 344 y Fu(=)d Ft(g)727 324 y Fp(\000)p Fo(1)725 357 y Fi(i)775 344 y Ft(g)798 351 y Fi(i)812 344 y Ft(ug)865 324 y Fp(\000)p Fo(1)863 357 y Fi(i)912 344 y Ft(g)935 351 y Fi(i)963 344 y Fu(=)h Ft(u)p Fu(,)h(so)h(that)h Ft(u)p Fu(\()p Ft(l)1299 351 y Fi(i)1312 344 y Fu(\))d(=)f Ft(l)1411 351 y Fi(i)1441 344 y Fu(for)j(all)f Ft(u)e Fr(2)h Ft(U)5 b Fu(,)228 405 y(hence)19 b Ft(l)382 412 y Fi(i)417 405 y Fr(2)i Ft(k)r Fu(.)34 b(F)l(or)20 b Ft(g)j Fu(:=)e Ft(g)779 412 y Fi(j)797 405 y Ft(g)822 384 y Fp(\000)p Fo(1)820 417 y Fi(i)890 405 y Fu(w)o(e)f(get)h Ft(g)r(g)1100 412 y Fi(i)1114 405 y Ft(U)26 b Fu(=)21 b Ft(g)1255 412 y Fi(j)1274 405 y Ft(U)26 b Fu(hence)19 b Ft(\033)r Fu(\()p Ft(g)r Fu(\)\()p Ft(i)p Fu(\))h(=)h Ft(j)228 465 y Fu(and)g Ft(u)355 472 y Fi(g)q(;i)418 465 y Fu(=)g Ft(g)502 444 y Fp(\000)p Fo(1)500 477 y Fi(j)549 465 y Ft(g)572 472 y Fi(j)591 465 y Ft(g)616 444 y Fp(\000)p Fo(1)614 477 y Fi(i)663 465 y Ft(g)686 472 y Fi(i)722 465 y Fu(=)g(id,)g(so)g(that)g Ft(l)1046 472 y Fi(i)1060 465 y Ft(e)1083 472 y Fi(j)1122 465 y Fu(=)g Ft(l)1196 472 y Fi(j)1214 465 y Ft(e)1237 472 y Fi(j)1255 465 y Fu(,)g(hence)f Ft(l)1445 472 y Fi(i)1479 465 y Fu(=)h Ft(l)1553 472 y Fi(j)1592 465 y Fu(for)g(all)228 525 y Ft(i;)8 b(j)s Fu(.)24 b(This)17 b(sho)o(ws)581 488 y Fh(P)642 525 y Ft(l)657 532 y Fi(i)671 525 y Ft(e)694 532 y Fi(i)723 525 y Fu(=)f Ft(\025)813 488 y Fh(P)874 525 y Ft(e)897 532 y Fi(i)927 525 y Fu(=)f Ft(\025)i Fr(2)e Ft(k)r Fu(.)25 b(Ob)o(viously)16 b(all)h(elemen)o(ts)e (of)i Ft(k)228 584 y Fu(remain)g(\014xed)h(under)h(the)f(action)h(of)g Ft(G)g Fu(so)g(that)g Ft(k)h Fu(=)e Ft(K)1320 565 y Fi(G)1350 584 y Fu(.)28 b(F)l(urthermore)17 b Ft(K)228 642 y Fu(is)g(separable)h (b)o(y)g(de\014nition.)25 b(T)l(o)19 b(sho)o(w)f(that)h Ft(G)f Fu(op)q(erates)h(strongly)f(distictly)228 700 y(it)f(su\016ces)g(to)h(\014nd)g(for)g(ev)o(ery)e Ft(g)i Fr(2)e Ft(G;)8 b(g)19 b Fr(6)p Fu(=)d(id)h(and)i Ft(e)1235 707 y Fi(i)1264 700 y Fr(2)e Ft(K)k Fu(and)e Ft(x)c Fr(2)i Ft(K)k Fu(suc)o(h)228 758 y(that)16 b Ft(g)r Fu(\()p Ft(x)p Fu(\))p Ft(e)447 765 y Fi(i)475 758 y Fr(6)p Fu(=)d Ft(xe)577 765 y Fi(i)591 758 y Fu(.)21 b(Assume)14 b(\014rst)j(that)f Ft(\033)r Fu(\()p Ft(g)r Fu(\)\()p Ft(i)p Fu(\))d Fr(6)p Fu(=)h Ft(i)p Fu(.)21 b(Cho)q(ose)c Ft(x)c Fu(=)h Ft(e)1562 765 y Fi(i)1576 758 y Fu(.)21 b(Then)228 816 y Ft(g)r Fu(\()p Ft(e)295 823 y Fi(i)309 816 y Fu(\))p Ft(e)351 823 y Fi(i)380 816 y Fu(=)15 b Ft(e)456 824 y Fi(\033)q Fo(\()p Fi(g)q Fo(\)\()p Fi(i)p Fo(\))564 816 y Ft(e)587 823 y Fi(i)616 816 y Fu(=)g(0)h Fr(6)p Fu(=)f Ft(e)785 823 y Fi(i)814 816 y Fu(=)g Ft(e)890 823 y Fi(i)904 816 y Ft(e)927 823 y Fi(i)941 816 y Fu(.)23 b(If)17 b Ft(\033)r Fu(\()p Ft(g)r Fu(\)\()p Ft(i)p Fu(\))d(=)i Ft(i)g Fu(then)h Ft(g)1414 796 y Fp(\000)p Fo(1)1412 829 y Fi(i)1462 816 y Ft(g)r(g)1510 823 y Fi(i)1540 816 y Fr(2)e Ft(U)23 b Fu(and)228 875 y Ft(u)18 b Fr(6)p Fu(=)h(id)g(since)f Ft(g)r(not)h Fu(=)g(id)o(.)30 b(Cho)q(ose)21 b(an)e Ft(l)h Fr(2)f Ft(L)g Fu(with)g Ft(u)p Fu(\()p Ft(l)q Fu(\))f Fr(6)p Fu(=)h Ft(l)h Fu(and)f Ft(x)g Fu(=)f Ft(l)q(e)1695 882 y Fi(i)1709 875 y Fu(.)228 933 y(Then)i Ft(g)r Fu(\()p Ft(x)p Fu(\))p Ft(e)473 940 y Fi(i)507 933 y Fu(=)g Ft(g)r Fu(\()p Ft(l)q(e)648 940 y Fi(i)661 933 y Fu(\))p Ft(e)703 940 y Fi(i)737 933 y Fu(=)h Ft(g)821 913 y Fp(\000)p Fo(1)819 946 y Fi(i)868 933 y Ft(g)r(g)916 940 y Fi(i)931 933 y Fu(\()p Ft(l)q Fu(\))p Ft(e)1008 940 y Fi(i)1021 933 y Ft(u)p Fu(\()p Ft(l)q Fu(\))p Ft(e)1126 940 y Fi(i)1159 933 y Fr(6)p Fu(=)f Ft(l)q(e)1256 940 y Fi(i)1290 933 y Fu(=)g Ft(l)q(e)1387 940 y Fi(i)1401 933 y Ft(e)1424 940 y Fi(i)1458 933 y Fu(=)g Ft(xe)1567 940 y Fi(i)1580 933 y Fu(.)33 b(This)228 991 y(concludes)16 b(the)g(pro)q(of.)1027 b Ff(\003)278 1125 y Fu(Observ)o(e)15 b(b)o(y)h(the)f(w)o(a)o(y)h(that) h Ft(k)r(C)883 1132 y Fo(2)919 1125 y Fu(has)g(no)f(non-trivial)g (forms,)f(since)h Ft(C)1616 1132 y Fo(2)1651 1125 y Fu(has)228 1183 y(trivial)g(automorphism)h(group,)h(so)h(the)f(corresp)q(onding)g (Galois)h(extension)e(of)228 1241 y(a)e(form)e(m)o(ust)h(b)q(e)g Ft(k)j Fu(itself.)j(Already)13 b(the)i(next)f(simplest)f(cases)h(after) h(studying)228 1299 y(the)h(forms)g(of)h Ft(k)r Fn(Z)p Fu(,)g Ft(k)r(C)657 1306 y Fo(3)677 1299 y Fu(,)f Ft(k)r(C)769 1306 y Fo(4)789 1299 y Fu(,)g(and)h Ft(k)r(C)976 1306 y Fo(6)1013 1299 y Fu(cause)g(unsatisfactory)g(calculations.)228 1357 y(W)l(e)f(discuss)g(the)g(case)g(of)h Fn(Q)p Ft(C)795 1364 y Fo(5)815 1357 y Fu(.)278 1415 y(The)k(automorphism)f(group)j(of) f Ft(C)951 1422 y Fo(5)992 1415 y Fu(is)f Ft(C)1081 1422 y Fo(4)1122 1415 y Fu(whic)o(h)g(has)h(exactly)f(one)g(non-)228 1474 y(trivial)c(subgroup)k Ft(C)625 1481 y Fo(2)644 1474 y Fu(.)30 b(The)19 b Ft(C)826 1481 y Fo(4)845 1474 y Fu(-Galois)h(extensions)f Ft(K)k Fu(of)c Fn(Q)g Fu(can)g(b)q(e)g(of)g (the)228 1532 y(follo)o(wing)d(forms)321 1621 y(1\))21 b Ft(K)f Fu(is)c(a)h Ft(C)571 1628 y Fo(4)591 1621 y Fu(-Galois)g(\014eld)e(extension)h(of)h Fn(Q)p Fu(,)321 1679 y(2\))k Ft(K)444 1665 y Fr(\030)444 1681 y Fu(=)497 1679 y Ft(L)11 b Fr(\002)g Ft(L)16 b Fu(where)g Ft(L)g Fu(is)g(a)h(quadratic)f(\014eld)g(extension)g(of)g Fn(Q)p Fu(,)321 1737 y(3\))21 b Ft(K)444 1723 y Fr(\030)444 1739 y Fu(=)497 1737 y Fn(Q)11 b Fr(\002)g Fn(Q)g Fr(\002)g Fn(Q)g Fr(\002)f Fn(Q)p Fu(.)278 1826 y(The)15 b(problem)e(is)i(no)o(w) g(to)h(describ)q(e)e(as)i(explicitly)c(as)k(p)q(ossible)f(all)g Ft(C)1571 1833 y Fo(2)1590 1826 y Fu(-)h(resp.)228 1884 y Ft(C)263 1891 y Fo(4)282 1884 y Fu(-Galois)j(\014eld)f(extensions)g Ft(K)k Fu(of)d Fn(Q)p Fu(,)f(to)h(describ)q(e)f(the)g(action)g(of)g Ft(C)1569 1891 y Fo(4)1607 1884 y Fu(on)h Ft(K)228 1942 y Fu(and)j(then)f(calculate)f(the)h(forms)g(according)g(to)h(Theorem)e (4.1.)37 b(Asso)q(ciated)228 2000 y(with)16 b(a)h Ft(C)415 2007 y Fo(4)434 2000 y Fu(-Galois)g(\014eld)f(extension)f Ft(K)21 b Fu(is)16 b(the)g(follo)o(wing)g(form)f(of)h Fn(Q)p Ft(C)1569 2007 y Fo(5)1589 2000 y Fu(:)385 2124 y Ft(H)443 2111 y Fr(\030)443 2126 y Fu(=)495 2124 y Fn(Q)p Fu([)p Ft(a)p Fu(])p Ft(=)p Fu(\()p Ft(a)660 2104 y Fo(5)690 2124 y Fr(\000)11 b Fu(5)p Ft(pa)814 2104 y Fo(3)845 2124 y Fu(+)g(\(5\()p Ft(p)980 2104 y Fo(2)1012 2124 y Fr(\000)f Fu(3)p Ft(q)r Fu(\))h Fr(\000)g Fu(10)1237 2079 y Fh(p)p 1288 2079 215 2 v 1288 2124 a Ft(q)r Fu(\()p Ft(p)1355 2110 y Fo(2)1385 2124 y Fr(\000)g Fu(4)p Ft(q)r Fu(\)\))p Ft(a)p Fu(\))228 2249 y(Here)22 b Ft(K)27 b Fu(is)c(the)g(splitting)g(\014eld)f(of)i Ft(x)968 2231 y Fo(4)1003 2249 y Fu(+)16 b Ft(px)1109 2231 y Fo(2)1144 2249 y Fu(+)g Ft(q)25 b Fu(and)1347 2207 y Fh(p)p 1396 2207 V 1396 2249 a Ft(q)r Fu(\()p Ft(p)1463 2235 y Fo(2)1494 2249 y Fr(\000)11 b Fu(4)p Ft(q)r Fu(\))25 b Fr(2)h Ft(k)228 2308 y Fu(necessarily)c(holds)i(if)f Ft(K)28 b Fu(is)23 b(a)h Ft(C)873 2315 y Fo(4)893 2308 y Fu(-Galois)g(\014eld)f (extension.)43 b(The)23 b(diagonal)228 2366 y(maps)15 b(can)i(b)q(e)f(describ)q(ed)g(b)o(y)228 2507 y(\001\()p Ft(a)p Fu(\))d(=)522 2473 y(1)p 403 2495 265 2 v 403 2541 a Ft(uv)r Fu(\()p Ft(u)504 2527 y Fo(2)533 2541 y Fr(\000)e Ft(v)609 2527 y Fo(2)628 2541 y Fu(\))647 2527 y Fo(2)672 2507 y Fu(\(\()p Ft(u)738 2486 y Fo(5)757 2507 y Fu(+)p Ft(v)821 2486 y Fo(5)840 2507 y Fu(\)\()p Ft(a)p Fr(\012)p Ft(a)p Fu(\))p Fr(\000)p Fu(\()p Ft(u)1074 2486 y Fo(3)1092 2507 y Fu(+)p Ft(v)1156 2486 y Fo(3)1175 2507 y Fu(\)\()p Ft(a)p Fr(\012)p Ft(b)p Fu(+)p Ft(b)p Fr(\012)p Ft(a)p Fu(\)+\()p Ft(u)p Fu(+)p Ft(v)r Fu(\)\()p Ft(b)p Fr(\012)p Ft(b)o Fu(\)\))p Ft(;)p eop %%Page: 16 16 16 15 bop 228 55 a Fl(16)545 b(BODO)13 b(P)m(AREIGIS)228 154 y Fu(where)386 295 y Ft(u)g Fu(=)479 194 y Fh(s)p 529 194 336 2 v 534 261 a Fr(\000)p Ft(p)e Fu(+)657 217 y Fh(p)p 707 217 153 2 v 44 x Ft(p)731 246 y Fo(2)762 261 y Fr(\000)g Fu(4)p Ft(q)p 534 283 326 2 v 685 329 a Fu(2)930 295 y(and)65 b Ft(v)16 b Fu(=)1164 194 y Fh(s)p 1214 194 337 2 v 1219 261 a Fr(\000)p Ft(p)11 b Fr(\000)1343 217 y Fh(p)p 1393 217 153 2 v 44 x Ft(p)1417 246 y Fo(2)1448 261 y Fr(\000)g Fu(4)p Ft(q)p 1219 283 327 2 v 1370 329 a Fu(2)1551 295 y Ft(:)278 414 y Fu(The)17 b(case)f(of)i Ft(K)599 400 y Fr(\030)599 416 y Fu(=)652 414 y Ft(L)12 b Fr(\002)f Ft(L)17 b Fu(with)g(quadratic)g(extension)f Ft(L=)p Fn(Q)i Fu(is)f(somewhat)228 472 y(easier.)k(W)l(e)16 b(get)277 562 y Ft(H)335 548 y Fr(\030)336 564 y Fu(=)388 562 y Fn(Q)p Fu([)p Ft(a)p Fu(])p Ft(=)p Fu(\()p Ft(a)553 541 y Fo(5)582 562 y Fu(+)11 b(5)p Ft(p)p Fu(\()p Ft(a)724 541 y Fo(3)755 562 y Fu(+)g Ft(pa)p Fu(\)\))66 b(with)e(\001\()p Ft(a)p Fu(\))13 b(=)h Ft(u)1315 541 y Fp(\000)p Fo(1)1362 562 y Fu(\(1)p Ft(;)8 b Fr(\000)p Fu(1\)\()p Ft(a)j Fr(\012)g Ft(a)p Fu(\))p Ft(:)228 651 y Fu(where)16 b Ft(L)e Fu(=)g Fn(Q)p Fu(\()p Ft(u)p Fu(\))i(is)g(the)g(splitting)f(\014eld)h(of)g Ft(x)1106 633 y Fo(2)1137 651 y Fr(\000)11 b Ft(p)p Fu(.)278 709 y(Finally)17 b(the)i(case)g(of)h Ft(K)762 695 y Fr(\030)762 711 y Fu(=)819 709 y Fn(Q)13 b Fr(\002)g Fn(Q)g Fr(\002)g Fn(Q)g Fr(\002)g Fn(Q)19 b Fu(leads)g(to)h(the)e(trivial)g(form)228 767 y Fn(Q)p Ft(C)305 774 y Fo(5)325 767 y Fu(.)278 825 y(The)f(other)g(simple)e(example)g(is)j(that)f(of)h(forms)e(of)i Fn(Q)p Ft(C)1335 832 y Fo(2)1366 825 y Fr(\002)12 b Ft(C)1452 832 y Fo(2)1471 825 y Fu(.)25 b(The)17 b(auto-)228 884 y(morphism)e(group)j(of)f Ft(C)687 891 y Fo(2)719 884 y Fr(\002)11 b Ft(C)804 891 y Fo(2)841 884 y Fu(is)17 b(the)g(symmetri)o(c)d(group)k Ft(S)1383 891 y Fo(3)1403 884 y Fu(.)24 b(No)o(w)17 b(w)o(e)g(ha)o(v)o(e)228 942 y(to)f(study)h(the)f(di\013eren)o(t)f(cases)i(of)f Ft(S)906 949 y Fo(3)926 942 y Fu(-Galois)h(extensions)321 1015 y(1\))k Ft(K)f Fu(is)c(an)h Ft(S)593 1022 y Fo(3)613 1015 y Fu(-Galois)g(\014eld)e(extension)h(of)h Fn(Q)p Fu(,)321 1074 y(2\))k Ft(K)444 1060 y Fr(\030)444 1076 y Fu(=)497 1074 y Ft(L)11 b Fr(\002)g Ft(L)16 b Fu(where)g Ft(L)g Fu(is)g(a)h Ft(C)955 1081 y Fo(3)975 1074 y Fu(-Galois)g (\014eld)e(extension)h(of)h Fn(Q)p Fu(,)321 1132 y(3\))k Ft(K)444 1118 y Fr(\030)444 1134 y Fu(=)497 1132 y Ft(L)11 b Fr(\002)g Ft(L)g Fr(\002)g Ft(L)16 b Fu(where)g Ft(L)h Fu(is)f(a)g(quadratic)g(\014eld)g(extension)g(of)g Fn(Q)p Fu(,)321 1190 y(4\))21 b Ft(K)444 1176 y Fr(\030)444 1192 y Fu(=)497 1190 y Fn(Q)11 b Fr(\002)g Fn(Q)g Fr(\002)g Fn(Q)g Fr(\002)f Fn(Q)i Fr(\002)e Fn(Q)i Fr(\002)e Fn(Q)p Fu(.)278 1264 y(In)16 b(the)g(\014rst)g(and)h(second)f(case)h(w)o(e)e (get)688 1353 y Ft(H)j Fu(=)c Fn(Q)p Fu([)p Ft(a)p Fu(])p Ft(=)p Fu(\()p Ft(a)p Fu(\()p Ft(a)1008 1333 y Fo(3)1037 1353 y Fu(+)d Ft(ua)f Fu(+)h Ft(v)r Fu(\)\))228 1443 y(where)j Ft(K)k Fu(is)d(the)f(splitting)f(\014eld)h(of)h Ft(x)932 1424 y Fo(3)959 1443 y Fu(+)7 b Ft(ux)g Fu(+)g Ft(v)16 b Fu(irreducible.)i(If)c Ft(D)i Fu(=)e Fr(\000)p Fu(4)p Ft(u)1657 1424 y Fo(3)1683 1443 y Fr(\000)228 1501 y Fu(27)p Ft(v)302 1483 y Fo(2)338 1501 y Fu(is)i(the)g (discriminan)o(t)e(then)249 1577 y(\001\()p Ft(a)p Fu(\))f(=)429 1558 y Fo(1)p 423 1566 30 2 v 423 1594 a Fi(D)458 1577 y Fu([)41 b Fr(\000)p Fu(2)p Ft(u)p Fu(\(3)p Ft(v)r(a)11 b Fr(\012)f Ft(a)h Fr(\000)g Ft(u)p Fu(\()p Ft(a)f Fr(\012)h Ft(c)g Fu(+)g Ft(c)g Fr(\012)g Ft(a)p Fu(\)\))513 1635 y(+)556 1616 y Fi(v)p 556 1624 19 2 v 556 1653 a Fo(2)580 1635 y Fu(\(4)p Ft(u)651 1617 y Fo(2)670 1635 y Ft(b)g Fr(\012)g Ft(b)g Fu(+)g(9)p Ft(c)g Fr(\012)g Ft(c)g Fu(+)g(6)p Ft(u)p Fu(\()p Ft(b)g Fr(\012)g Ft(c)g Fu(+)g Ft(c)g Fr(\012)g Ft(b)p Fu(\)\))g Fr(\000)f Fu(9)p Ft(v)1505 1617 y Fo(2)1525 1635 y Fu(\()p Ft(a)h Fr(\012)f Ft(b)h Fu(+)g Ft(b)g Fr(\012)g Ft(a)p Fu(\)])228 1723 y(where)775 1804 y Ft(b)j Fu(=)867 1770 y(4)p 867 1792 26 2 v 867 1838 a Ft(v)897 1804 y(a)923 1783 y Fo(3)953 1804 y Fu(+)1007 1770 y(4)p Ft(u)p 1007 1792 53 2 v 1021 1838 a(v)1065 1804 y(a)c Fu(+)h(3)228 1898 y(and)646 1986 y Ft(c)j Fu(=)g Fr(\000)777 1953 y Fu(4)p Ft(u)p 777 1975 V 790 2020 a(v)834 1986 y(a)860 1966 y Fo(3)890 1986 y Fu(+)d(2)p Ft(a)989 1966 y Fo(2)1020 1986 y Fr(\000)1075 1953 y Fu(4)p Ft(u)1127 1935 y Fo(2)p 1075 1975 72 2 v 1098 2020 a Ft(v)1152 1986 y(a)f Fr(\000)h Fu(4)p Ft(u:)278 2080 y Fu(In)16 b(the)g(third)g(case)g(of)g Ft(K)762 2067 y Fr(\030)763 2082 y Fu(=)815 2080 y Ft(L)11 b Fr(\002)g Ft(L)g Fr(\002)g Ft(L)16 b Fu(w)o(e)g(get)671 2170 y Ft(H)i Fu(=)c Fn(Q)p Fu([)p Ft(a)p Fu(])p Ft(=)p Fu(\(\()p Ft(a)965 2149 y Fo(2)995 2170 y Fr(\000)c Fu(1\)\()p Ft(a)1132 2149 y Fo(2)1163 2170 y Fr(\000)h Ft(u)p Fu(\)\))228 2259 y(where)16 b Ft(L)g Fu(is)g(the)g(splitting)g(\014eld)g(of)g Ft(x)933 2241 y Fo(2)964 2259 y Fr(\000)10 b Ft(u)16 b Fu(and)h(the)f(diagonal)h(is)297 2370 y(\001\()p Ft(a)p Fu(\))c(=)528 2337 y(1)p 472 2359 137 2 v 472 2405 a Ft(u)500 2390 y Fo(2)531 2405 y Fr(\000)e Ft(u)622 2330 y Fh(\002)642 2370 y Fu(\()p Ft(u)689 2350 y Fo(2)720 2370 y Fr(\000)g Ft(u)g Fr(\000)f Fu(1\))p Ft(a)h Fr(\012)g Ft(a)g Fr(\000)g Ft(a)1101 2350 y Fo(3)1131 2370 y Fr(\012)g Ft(a)1207 2350 y Fo(3)1237 2370 y Fu(+)g Ft(a)1312 2350 y Fo(3)1343 2370 y Fr(\012)f Ft(a)h Fu(+)g Ft(a)g Fr(\012)g Ft(a)1591 2350 y Fo(3)1610 2330 y Fh(\003)1639 2370 y Ft(:)278 2479 y Fu(The)16 b(case)f(of)i Ft(K)595 2465 y Fr(\030)595 2481 y Fu(=)647 2479 y Fn(Q)11 b Fr(\002)f Fn(Q)g Fr(\002)h Fn(Q)f Fr(\002)g Fn(Q)h Fr(\002)f Fn(Q)h Fr(\002)f Fn(Q)16 b Fu(leads)g(to)g(the)g(trivial)e(form)228 2537 y Fn(Q)p Fu(\()p Ft(C)324 2544 y Fo(2)355 2537 y Fr(\002)c Ft(C)439 2544 y Fo(2)459 2537 y Fu(\).)p eop %%Page: 17 17 17 16 bop 465 55 a Fl(F)o(ORMS)18 b(OF)e(HOPF)h(ALGEBRAS)g(AND)g (GALOIS)f(THEOR)m(Y)199 b(17)228 154 y Fn(Problems)17 b(4.3.)j Fu(The)d(generators)h(of)f(Hopf)f(algebra)h(forms)f(and)i (their)e(diag-)228 212 y(onals)i(are)g(rather)h(arbitrary)l(.)26 b(It)17 b(often)h(turns)h(out)f(that)h(either)e(the)g(diagonal)228 270 y(or)i(the)g(ideal)f(to)h(b)q(e)g(factored)g(out)g(can)g(b)q(e)g(c) o(hosen)f(to)i(b)q(e)f(relativ)o(ely)d(simple,)228 329 y(but)i(not)h(b)q(oth.)29 b(Is)18 b(there)g(a)h(canonical)f(c)o(hoice)f (of)i(the)f(generators)h(of)f(a)h(Hopf)228 387 y(algebra)g(form?)554 367 y Fo(1)601 387 y Fu(Is)g(there)g(a)g(w)o(a)o(y)g(to)h(determine)c (the)j(minimal)d(n)o(um)o(b)q(er)h(of)228 445 y(generators?)30 b(Can)20 b(one)f(describ)q(e)f(the)h("cyclic")e(Hopf)i(algebra)g (forms?)29 b(This)228 503 y(seems)15 b(to)i(b)q(e)g(of)h(in)o(terest)d (for)i(the)g(represen)o(tation)f(theory)h(of)g(Hopf)g(algebras,)228 561 y(lik)o(e)d(cyclic)g(groups)k(are)e(for)h(the)f(represen)o(tation)f (theory)h(of)h(groups.)278 630 y(Another)g(problem)f(area)i(arises)f (from)g(the)g(follo)o(wing)g(considerations.)25 b(Let)228 688 y Ft(k)r(C)290 695 y Fi(n)332 688 y Fu(b)q(e)19 b(the)f(group)h (algebra)h(of)e(a)h(\014nite)f(cyclic)f(group)i(and)h(let)d Ft(k)k Fu(b)q(e)e(a)g(\014eld)228 747 y(with)d(c)o(har\()p Ft(k)r Fu(\))24 b Ft(=)-31 b Fr(\000)10 b Ft(n)p Fu(.)22 b(Then)16 b(the)f(group)i(algebra)g(is)e(semisimple)d(b)o(y)k(Masc)o (hk)o(e's)228 805 y(theorem.)24 b(Let)18 b Ft(K)j Fu(b)q(e)d(a)g (\014eld)f(extension)h(of)g Ft(k)h Fu(or)f(a)g(comm)o(utativ)o(e)c (separable)228 863 y(algebra.)37 b(Then)22 b(ev)o(ery)e Ft(K)t Fu(-form)h Ft(H)26 b Fu(of)c Ft(k)r(C)1071 870 y Fi(n)1116 863 y Fu(is)f(again)i(semisim)o(ple)o(,)d(since)h(a)228 921 y(nilp)q(oten)o(t)15 b(ideal)g(of)h Ft(H)j Fu(w)o(ould)d(remain)e (nilp)q(oten)o(t)h(in)g Ft(K)f Fr(\012)9 b Ft(H)1395 907 y Fr(\030)1396 923 y Fu(=)1448 921 y Ft(K)t(C)1528 928 y Fi(n)1567 921 y Fu(in)15 b(b)q(oth)228 979 y(cases,)24 b(but)f Ft(K)t(C)548 986 y Fi(n)594 979 y Fu(is)g(still)f(semisim)o (ple)o(.)38 b(There)23 b(ma)o(y)e(b)q(e)i(forms)f(whic)o(h)g(are)228 1037 y(ev)o(en)14 b(b)q(etter)h(in)g(their)g(represen)o(tation)f(prop)q (erties)i(as)g(the)f(example)e(of)i(part)h(I.)228 1095 y(sho)o(ws.)278 1153 y(The)h(Hopf)f(algebra)i Ft(H)h Fu(=)c Fn(R)p Fu([)p Ft(c;)8 b(s)p Fu(])p Ft(=)p Fu(\()p Ft(c)982 1135 y Fo(2)1012 1153 y Fu(+)k Ft(s)1085 1135 y Fo(2)1116 1153 y Fr(\000)f Fu(1)p Ft(;)d(cs)p Fu(\))17 b(is)g(a)g Fn(C)p Fu(-form)g(of)g Fn(R)p Ft(C)1689 1160 y Fo(4)1709 1153 y Fu(.)228 1211 y(It)g(is)g(easy)g(to)h(see)e(that)i Ft(H)747 1198 y Fr(\030)747 1214 y Fu(=)801 1211 y Fn(R)12 b Fr(\002)f Fn(R)h Fr(\002)f Fn(R)h Fr(\002)f Fn(R)17 b Fu(as)h Fn(R)p Fu(-algebras.)25 b(Th)o(us)18 b Ft(H)j Fu(is)228 1270 y(absolutely)c(semisimpl)o(e,)e(i.e.)24 b(all)18 b(its)f(simple)f(mo)q(dules)h(are)h(one-dimensional)228 1328 y(o)o(v)o(er)i(the)i(base)f(\014eld.)37 b Fn(R)p Ft(C)758 1335 y Fo(4)777 1328 y Fu(,)22 b(ho)o(w)o(ev)o(er,)g(is)f(not) h(absolutely)f(semisim)o(ple.)34 b(It)228 1386 y(decomp)q(oses)22 b(as)h Fn(R)p Ft(C)642 1393 y Fo(4)686 1372 y Fr(\030)687 1388 y Fu(=)750 1386 y Fn(R)15 b Fr(\002)g Fn(R)g Fr(\002)g Fn(C)24 b Fu(as)f(an)g(algebra,)h(so)g(it)e(has)h(a)g(t)o(w)o(o-)228 1444 y(dimensional)14 b(simple)g(mo)q(dule.)278 1502 y(So)20 b(there)g(is)g(the)g(problem)e(of)j(determining)d(whic)o(h)h (group)i(algebras)g(ha)o(v)o(e)228 1560 y(absolutly)f(semisimpl)o(e)d (forms)j(and)h(to)g(describ)q(e)e(all)h(those)h(forms.)33 b(If)20 b(ev)o(ery)228 1618 y(semisim)o(ple)9 b(group)j(algebra)h(had)f (an)g(absolutely)g(semisimpl)o(e)d(form)h(this)i(w)o(ould)228 1676 y(mean,)i(that)i(one)g(do)q(es)h(not)f(need)g(to)g(extend)f(the)g (base)i(\014eld)e(of)h(a)g(group)h(alge-)228 1734 y(bra)g Ft(k)r(G)h Fu(to)f(obtain)g(total)h(splitting,)e(but)h(that)g(the)g (splitting)f(can)h(already)g(b)q(e)228 1793 y(obtained)12 b(o)o(v)o(er)e(the)i(base)g(ring)g(for)g(a)g(suitable)f(form)f Ft(H)t Fu(.)20 b(Since)11 b(w)o(e)g(are)h(not)g(talk-)228 1851 y(ing)18 b(ab)q(out)h(algebra)f(forms)f(but)h(ab)q(out)h(Hopf)f (algebra)g(forms)f(the)g(p)q(ossibilit)o(y)228 1909 y(of)g(tensoring)g Ft(H)t Fu(-mo)q(dules)g(o)o(v)o(er)f(the)g(base)i(\014eld)e(|)h(an)g (imp)q(ortan)o(t)f(tec)o(hnique)228 1967 y(for)g(represen)o(tation)g (theory)g(|)g(is)g(preserv)o(ed.)228 2036 y Fn(Theorem)24 b(4.4.)f Fq(If)g Ft(k)i Fq(is)f(a)f(\014eld)i(of)e(char)n(acteristic)g (not)h(dividing)g Ft(n)p Fq(,)h(then)228 2094 y(the)18 b(Hopf)f(algebr)n(a)h Ft(k)r(C)654 2101 y Fi(n)695 2094 y Fq(has)f(a)g(uniquely)i(determine)n(d)f(absolutely)h(semisimple)228 2153 y(Hopf)e(algebr)n(a)h(form)f Ft(k)656 2134 y Fi(C)681 2138 y Fd(n)718 2153 y Fu(=)d(\()p Ft(k)r(C)851 2160 y Fi(n)874 2153 y Fu(\))893 2134 y Fp(\003)913 2153 y Fq(.)228 2238 y(Pr)n(o)n(of.)19 b Fu(An)o(y)d(absolutely)h(semisim)o (ple)c(form)j(of)h Ft(k)r(C)1196 2245 y Fi(n)1237 2238 y Fu(has)h(underlying)e(algebra)228 2296 y Ft(k)255 2278 y Fi(I)275 2296 y Fu(.)33 b(But)20 b Ft(k)450 2278 y Fi(I)490 2296 y Fu(is)g(a)h(Hopf)f(algebra)h(i\013)f Ft(I)j Fu(is)d(a)h(\014nite)f(group.)34 b(After)19 b(base)i(\014eld)228 2354 y(extension)14 b(the)h(group)h(structure)f(of)h Ft(I)i Fu(remains)c(unc)o(hanged,)h(so)h(there)f(can)g(b)q(e)p 228 2388 250 2 v 278 2421 a Fo(1)295 2437 y Fv(Added)f(in)f(pro)q(of:)k (A)d(complete)e(answ)o(er)i(to)f(this)g(question)h(for)f(the)g(case)i (of)d(group)h(rings)228 2487 y(has)h(b)q(een)g(giv)o(en)f(recen)o(tly)i (b)o(y)e(the)h(author)g(in)f Fm(Twiste)n(d)h(gr)n(oup)g(rings)f Fv(submitted)g(to)h(Comm.)228 2537 y(Alg.)p eop %%Page: 18 18 18 17 bop 228 55 a Fl(18)545 b(BODO)13 b(P)m(AREIGIS)228 154 y Fu(at)j(most)f(one)h(group)h(structure)e(on)h Ft(I)k Fu(and)c(at)g(most)f(one)h(Hopf)g(algebra)g(struc-)228 212 y(ture)j(on)h Ft(k)432 194 y Fi(I)472 212 y Fu(so)g(that)g Ft(k)671 194 y Fi(I)710 212 y Fu(is)g(a)f(form)g(of)h Ft(k)r(C)1046 219 y Fi(n)1069 212 y Fu(.)31 b(So)20 b(an)g(absolutely)f (semisimple)228 270 y(form)h(of)i Ft(k)r(C)471 277 y Fi(n)516 270 y Fu(is)f(a)h(Hopf)f(algebra)h Ft(k)943 252 y Fi(G)994 270 y Fu(with)f(a)h(uniquely)e(determined)f(com-)228 329 y(m)o(utativ)o(e)c(group)k Ft(G)g Fu(of)f(order)g Ft(n)p Fu(.)26 b(W)l(e)18 b(sho)o(w)h Ft(G)1150 315 y Fr(\030)1150 331 y Fu(=)1206 329 y Ft(C)1241 336 y Fi(n)1282 329 y Fu(so)g(that)f Ft(k)1478 310 y Fi(C)1503 314 y Fd(n)1544 329 y Fu(b)q(ecomes)228 387 y(the)f(absolutely)g(semisimpl)o (e)e(form)h(of)i Ft(k)r(C)1030 394 y Fi(n)1053 387 y Fu(.)25 b(It)17 b(su\016ces)g(to)h(sho)o(w)g(this)f(o)o(v)o(er)g(a)228 445 y(\014eld)f Ft(k)i Fu(con)o(taining)e(an)h Ft(n)p Fu(-th)g(primitiv)n(e)c(ro)q(ot)18 b(of)e(unit)o(y)l(.)21 b(But)16 b(then)g Ft(k)r(C)1571 452 y Fi(n)1611 445 y Fu(splits)228 503 y(completely)d(and)k(the)f(statemen)o(t)e(is)i(w)o (ell)f(kno)o(wn.)505 b Ff(\003)278 607 y Fu(This)21 b(unique)g (absolutely)g(semisimpl)o(e)e(form)h(of)i Ft(k)r(C)1291 614 y Fi(n)1336 607 y Fu(is)f(asso)q(ciated)h(with)228 665 y(an)e Ft(F)7 b Fu(-Galois)20 b(extension)f Ft(K)24 b Fu(of)d Ft(k)h Fu(with)d Ft(F)1071 651 y Fr(\030)1071 667 y Fu(=)1130 665 y(Aut)o(\()p Ft(C)1266 672 y Fi(n)1290 665 y Fu(\).)32 b(It)19 b(turns)h(out)h(that)228 723 y Ft(k)r Fu([)p Ft(x)p Fu(])p Ft(=)p Fu(\()p Ft(')386 730 y Fi(n)408 723 y Fu(\()p Ft(x)p Fu(\)\))h(is)f(an)h Ft(F)7 b Fu(-Galois)21 b(extension)g(and)h(asso)q(ciated)h(to)f Ft(k)1502 705 y Fi(C)1527 709 y Fd(n)1550 723 y Fu(.)37 b Ft(')1633 730 y Fi(n)1657 723 y Fu(\()p Ft(x)p Fu(\))228 781 y(is)22 b(the)g Ft(n)p Fu(-th)h(cyclotomic)c(p)q(olynomial.)39 b(In)22 b(general)g Ft(k)r Fu([)p Ft(x)p Fu(])p Ft(=)p Fu(\()p Ft(')1424 788 y Fi(n)1447 781 y Fu(\()p Ft(x)p Fu(\)\))g(will)f(not)228 839 y(b)q(e)g(a)f(\014eld)g(extension)g(of)h Ft(k)r Fu(.)35 b(According)20 b(to)h(Theorem)e(4.2)i(and)g(with)f(some) 228 897 y(additional)e(calculations)g(one)h(can)g(see)f(that)h Ft(k)r Fu([)p Ft(x)p Fu(])p Ft(=)p Fu(\()p Ft(')1259 904 y Fi(n)1281 897 y Fu(\()p Ft(x)p Fu(\)\))1384 883 y Fr(\030)1384 899 y Fu(=)1440 897 y Ft(k)r Fu(\()p Ft(\020)1507 904 y Fi(n)1531 897 y Fu(\))12 b Fr(\002)h Ft(:)8 b(:)g(:)j Fr(\002)228 955 y Ft(k)r Fu(\()p Ft(\020)295 962 y Fi(n)319 955 y Fu(\).)228 1032 y Fn(Problems)16 b(4.5.)k Fu(It)c(w)o(ould)g(b)q (e)g(in)o(teresting)f(to)h(kno)o(w)g(whic)o(h)g(group)h(algebras)228 1090 y(o)o(v)o(er)i Fn(Q)i Fu(ha)o(v)o(e)f(absolutely)g(semisimpl)o(e)e (forms.)33 b(The)21 b(Hopf)f(algebra)h Fn(Q)p Ft(S)1645 1097 y Fi(n)1689 1090 y Fu(is)228 1148 y(itself)10 b(absolutely)h (semisim)o(ple.)16 b(There)11 b(are)g(also)h(examples)d(of)j(groups)g Ft(G)g Fu(whose)228 1207 y(group)17 b(algebras)g(ha)o(v)o(e)e(no)i (absolutely)f(semisimpl)o(e)d(forms.)492 1354 y(5.)43 b Fs(Sep)m(arable)17 b(Hopf)g(Galois)h(Extensions)228 1441 y Fn(Problems)j(5.1.)h Fu(In)d(part)i(I)q(I)q(I)e(w)o(e)h(ha)o(v)o (e)f(seen)h(examples)e(of)i(separable)g(\014eld)228 1499 y(extensions)d Ft(K)q(=k)j Fu(whic)o(h)d(are)h(Hopf)f(Galois.)26 b(All)16 b(the)i(examples)d(w)o(ere)i(in)g(fact)228 1557 y("almost)12 b(classically")f(Galois.)20 b(A)12 b(16-dimensional)f (example)f(of)j(a)g(Hopf)f(Galois)228 1615 y(extension)k(whic)o(h)f(is) h(not)h("almost)f(classically")f(Galois)i(is)f(giv)o(en)g(in)g([GP].)f (M.)228 1674 y(T)l(ak)o(euc)o(hi)d(has)i(c)o(hec)o(k)o(ed)d(that)j(all) f(that)h(all)f(Hopf)h(Galois)g(extensions)f(of)h(dimen-)228 1732 y(sion)j(less)g(than)h(8)g(are)f("almost)g(classically")f(Galois.) 25 b(The)17 b(ob)o(vious)g(question)228 1790 y(is,)f(are)h(there)f (prop)q(er)h(Hopf)f(Galois)i(extensions)e(of)h(dimension)e(less)h(than) h(16?)228 1848 y(Questions)h(ab)q(out)h(the)e(corresp)q(ondence)h(b)q (et)o(w)o(een)e("normal")i(Hopf)g(subalge-)228 1906 y(bras)h(and)g (Hopf)f(Galois)h(sub\014elds)g(ha)o(v)o(e)f(b)q(een)g(addresses)h(in)f ([C2].)28 b(Man)o(y)18 b(of)228 1964 y(those)j(questions)g(are)h(still) e(op)q(en.)36 b(Childs)21 b(also)g(addresses)h(the)f(question)g(of)228 2022 y(the)f(uniqueness)g(of)h(the)f(Hopf)h(algebra)g Ft(H)j Fu(w.r.t.)34 b(whic)o(h)19 b(a)i(separable)g(\014eld)228 2080 y(extension)15 b(is)g(Hopf)g(Galois.)21 b(He)15 b(obtains)h(results)f(for)h("classical")f(Galois)h(\014eld)228 2138 y(extensions.)24 b(He)16 b(sho)o(ws)j(that)e(the)g(Hopf)h(algebra) f Ft(H)22 b Fu(is)17 b(nev)o(er)f(unique)h(if)g Ft(G)g Fu(is)228 2197 y(cyclic)c(of)j(o)q(dd)h(prime)c(p)q(o)o(w)o(er)j (order.)21 b Ft(H)f Fu(is)15 b(nev)o(er)g(unique)g(for)h(non-ab)q (elian)g Ft(G)p Fu(.)228 2255 y(This)i(needs)f(a)i(di\013eren)o(t)e (pro)q(of,)i(ho)o(w)o(ev)o(er,)d(than)j(giv)o(en)e(in)g([C2].)26 b(Childs)18 b(also)228 2313 y(sho)o(ws)e(that)g Ft(H)k Fu(is)c(unique)f(if)g Ft(G)h Fu(is)g(cyclic)d(of)j(prime)e(order,)h(a)h (result)g(whic)o(h)f(w)o(e)228 2371 y(will)g(extend)g(b)q(elo)o(w.)278 2448 y(Assume)d(that)i(w)o(e)g(ha)o(v)o(e)f(the)h(same)f(setup)h Ft(K)q(=k)r(;)1198 2435 y Fu(~)1184 2448 y Ft(K)5 b(=k)r(;)j(G;)g(G) 1401 2430 y Fp(0)1413 2448 y Ft(;)g(S;)g(B)17 b Fu(as)e(in)e(I)q(I)q (I.)228 2506 y(In)j([C2])g(the)g(follo)o(wing)g(result)f(is)h(sho)o (wn.)p eop %%Page: 19 19 19 18 bop 465 55 a Fl(F)o(ORMS)18 b(OF)e(HOPF)h(ALGEBRAS)g(AND)g (GALOIS)f(THEOR)m(Y)199 b(19)228 154 y Fn(Prop)r(osition)19 b(5.2.)h Ft(G)f Fq(normalizes)g(the)g(r)n(e)n(gular)f(sub)n(gr)n(oup)f Ft(N)24 b Fq(of)18 b Ft(B)j Fq(i\013)d Ft(G)h Fq(is)228 212 y(a)e(sub)n(gr)n(oup)g(of)g(the)h(holomorph)f Fu(Hol)o(\()p Ft(N)5 b Fu(\))15 b(=)e Ft(N)p 1132 212 3 21 v 17 w Fr(\002)e Fu(Aut)o(\()p Ft(N)5 b Fu(\))p Fq(.)278 284 y Fu(W)l(e)16 b(extend)f(Theorem)g(2)i(of)f([C2])g(as)h(follo)o(ws:)228 356 y Fn(Theorem)j(5.3.)h Fq(L)n(et)f Ft(K)q(=k)j Fq(b)n(e)e(a)f(sep)n (ar)n(able)g(\014eld)i(extension)g(of)e(de)n(gr)n(e)n(e)g Fu([)p Ft(K)j Fu(:)228 414 y Ft(k)r Fu(])13 b(=)h Ft(p)k Fq(a)f(prime.)22 b(The)c(fol)r(lowing)i(ar)n(e)c(e)n(quivalent:)321 486 y Fu(1\))21 b Ft(K)q(=k)f Fq(is)e(Hopf)f(Galois.)321 544 y Fu(2\))k Ft(K)q(=k)f Fq(is)e(almost)f(classic)n(al)r(ly)i (Galois.)321 603 y Fu(3\))i Ft(G)d Fq(is)f(solvable.)278 674 y(If)g(any)h(\(and)h(al)r(l\))g(of)f(these)h(c)n(onditions)f(hold)h (then)g(the)f(Hopf)g(algebr)n(a)h Ft(H)j Fq(is)228 733 y(unique)d(for)e Ft(K)q(=k)j(H)t Fq(-Galois.)228 825 y(Pr)n(o)n(of.)f Fu(If)i Ft(K)q(=k)j Fu(is)c Ft(H)t Fu(-Galois)j(then)e (there)f(is)h(a)h(regular)f(subgroup)h Ft(N)27 b Fu(of)21 b Ft(S)1702 832 y Fi(p)228 883 y Fu(suc)o(h)f(that)h Ft(G)h Fr(\022)g Fu(Hol)o(\()p Ft(N)5 b Fu(\))22 b(=)f Ft(N)p 871 883 V 19 w Fr(\002)14 b Fu(Aut\()p Ft(N)5 b Fu(\),)21 b(the)g(holomorph)f(of)h Ft(N)5 b Fu(.)35 b(Since)228 941 y Ft(N)286 927 y Fr(\030)286 943 y Fu(=)338 941 y Ft(C)373 948 y Fi(p)409 941 y Fu(the)16 b(holomorph)g(Hol\()p Ft(N)5 b Fu(\))908 927 y Fr(\030)908 943 y Fu(=)960 941 y Ft(C)995 948 y Fi(p)p 1031 941 V 1026 941 a Fr(\002)11 b Ft(C)1111 948 y Fi(p)p Fp(\000)p Fo(1)1176 941 y Fu(,)k(hence)h Ft(G)h Fu(is)f(solv)m(able.)278 999 y(Let)j Ft(G)g Fu(b)q(e)g(solv)m (able.)29 b(Since)17 b Ft(K)23 b Fu(=)18 b Ft(k)r Fu(\()p Ft(a)p Fu(\))g(with)h Ft(a)f Fu(a)h(zero)g(of)g(an)g(irreducible)228 1057 y(p)q(olynomial)j Ft(f)29 b Fu(of)24 b(degree)f Ft(p)p Fu(,)i Ft(G)f Fu(is)g(a)g(subgroup)h(of)e Ft(S)1303 1064 y Fi(p)1323 1057 y Fu(,)i(hence)e Ft(p=)p Fr(j)p Ft(G)p Fr(j)i Fu(and)228 1115 y Ft(p)252 1097 y Fo(2)296 1115 y Ft(=)-30 b Fr(\000)13 b(j)p Ft(G)p Fr(j)p Fu(.)25 b(Belo)o(w)16 b(w)o(e)h(sho)o(w)h(that)g(for)g(a)g(solv)m(able)f(group) i Ft(G)f Fu(there)f(is)g(a)h(c)o(hain)228 1173 y(of)e(normal)g (subgroups)i(of)e Ft(G)h Fu(\(!\))703 1259 y Ft(e)10 b(/)i(:)c(:)g(:)i(/)i(G)914 1266 y Fo(2)945 1259 y Ft(/)f(G)1018 1266 y Fo(1)1050 1259 y Ft(/)g(G)1123 1266 y Fo(0)1157 1259 y Fu(=)j Ft(G)228 1345 y Fu(with)i Ft(G)377 1352 y Fi(i)391 1345 y Ft(=G)453 1352 y Fi(i)p Fo(+1)527 1331 y Fr(\030)528 1347 y Fu(=)580 1345 y(\()p Fn(Z)p Ft(=p)681 1352 y Fi(i)696 1345 y Fn(Z)p Fu(\))749 1327 y Fi(e)765 1332 y Fd(i)781 1345 y Fu(.)21 b(Consider)c(the)f(sequence)f(of)h (sub\014elds)579 1435 y Ft(k)g Fu(=)d Ft(k)696 1442 y Fo(0)730 1435 y Fr(\032)h Ft(k)808 1442 y Fo(1)842 1435 y Fr(\032)f Ft(:)8 b(:)g(:)14 b Fr(\032)f Ft(k)1043 1442 y Fi(m)p Fp(\000)p Fo(1)1136 1435 y Fr(\032)g Ft(k)1213 1442 y Fi(m)1261 1435 y Fu(=)1326 1422 y(~)1313 1435 y Ft(K)t(;)228 1521 y Fu(with)21 b Ft(k)369 1528 y Fi(i)p Fo(+1)428 1521 y Ft(=k)477 1528 y Fi(i)513 1521 y Fu(Galois)h(with)e (Galois)i(group)g(\()p Fn(Z)p Ft(=p)1182 1528 y Fi(i)1197 1521 y Fn(Z)p Fu(\))1250 1503 y Fi(e)1266 1508 y Fd(i)1303 1521 y Fu(and)g Ft(k)1428 1528 y Fi(i)p Fo(+1)1487 1521 y Ft(=k)i Fu(normal.)228 1583 y(W)l(e)f(get)g Ft(a)j Fr(2)g Ft(k)543 1590 y Fi(m)599 1583 y Fu(and)e Ft(a)31 b(=)-29 b Fr(2)26 b Ft(k)837 1590 y Fi(m)p Fp(\000)p Fo(1)915 1583 y Fu(,)f(otherwise)1191 1570 y(~)1178 1583 y Ft(K)30 b Fr(\022)25 b Ft(k)1338 1590 y Fi(m)p Fp(\000)p Fo(1)1441 1583 y Fu(since)d Ft(k)1592 1590 y Fi(m)p Fp(\000)p Fo(1)1671 1583 y Ft(=k)228 1641 y Fu(normal.)i(Then)17 b(the)g(minimal)d(p)q(olynomial)j Ft(f)22 b Fu(of)c Ft(a)f Fu(is)h(irreducible)d(o)o(v)o(er)h Ft(k)1643 1648 y Fi(m)p Fp(\000)p Fo(1)228 1699 y Fu(b)o(y)j(the)h(last)h(lemm)o(a.)30 b(So)21 b(all)e(the)h Ft(k)r Fu(-generating)h(elemen)o(ts)c(1)p Ft(;)8 b(a;)g(:)g(:)g(:)f(;)h(a)1598 1681 y Fi(p)p Fp(\000)p Fo(1)1683 1699 y Fu(of)228 1757 y Ft(K)22 b Fu(are)17 b(linearly)f(indep)q(enden)o(t)h(o)o(v)o(er)g Ft(k)958 1764 y Fi(m)p Fp(\000)p Fo(1)1036 1757 y Fu(.)26 b(Th)o(us)18 b Ft(K)j Fu(and)e Ft(k)1385 1764 y Fi(m)p Fp(\000)p Fo(1)1481 1757 y Fu(are)f(linearly)228 1818 y(disjoin)o(t)f(and)h Ft(p)523 1825 y Fi(m)573 1818 y Fu(=)e Ft(p)p Fu(.)25 b(Since)17 b Ft(p)843 1800 y Fo(2)888 1818 y Ft(=)-30 b Fr(\000)12 b Fu([)960 1805 y(~)947 1818 y Ft(K)20 b Fu(:)15 b Ft(k)r Fu(])j(and)g Ft(p)e Fu(=)g Ft(p)1310 1825 y Fi(m)1360 1818 y Fu(=)g([)p Ft(k)1453 1825 y Fi(m)1502 1818 y Fu(:)g Ft(k)1557 1825 y Fi(m)p Fp(\000)p Fo(1)1636 1818 y Fu(])h(w)o(e)228 1880 y(get)j Ft(k)338 1887 y Fi(m)p Fp(\000)p Fo(1)431 1880 y Fr(\012)13 b Ft(K)549 1866 y Fr(\030)549 1882 y Fu(=)608 1880 y Ft(k)633 1887 y Fi(m)p Fp(\000)p Fo(1)726 1880 y Fr(\001)h Ft(K)24 b Fu(=)d Ft(k)903 1887 y Fi(m)958 1880 y Fu(=)1029 1868 y(~)1016 1880 y Ft(K)t Fu(.)34 b(So)20 b(b)o(y)g(Theorem)f(3.2)i(the)f (\014eld)228 1938 y(extension)15 b Ft(K)q(=k)20 b Fu(is)c(almost)f (classically)g(Galois.)278 1997 y(T)l(o)f(see)f(that)g(the)h(Hopf)f (algebra)h Ft(H)j Fu(together)d(with)f(the)g(Galois)h(op)q(eration)g (is)228 2055 y(uniquely)g(determined,)f(observ)o(e)i(that)h Ft(p=)p Fr(j)p Ft(G)p Fr(j)h Fu(and)f Ft(G)e Fr(\022)g Ft(N)p 1361 2055 V 15 w Fr(\002)9 b Fu(Aut\()p Ft(N)c Fu(\))16 b(and)g Ft(N)228 2113 y Fu(the)d(only)g(Sylo)o(w)h(p-subgroup) h(of)f(Hol)o(\()p Ft(N)5 b Fu(\))14 b(imply)d(that)j(the)f(Sylo)o(w)g (p-subgroup)228 2171 y(of)19 b Ft(G)h Fu(is)e Ft(N)5 b Fu(,)20 b(whic)o(h)e(is)h(unique.)29 b(Th)o(us)19 b Ft(G)g Fu(=)g Ft(N)p 1153 2171 V 17 w Fr(\002)13 b Ft(A)19 b Fu(with)g(a)g(subgroup)h Ft(A)e Fr(\022)228 2229 y Fu(Aut)o(\()p Ft(N)5 b Fu(\).)24 b(Consequen)o(tly)16 b Ft(N)21 b Fu(=)15 b Ft(G)884 2236 y Fi(p)919 2229 y Fr(\022)g Ft(B)k Fu(is)e(uniquely)f(determined)e(and)k(so)f(is)228 2287 y Ft(H)j Fu(b)o(y)c(Theorem)f(3.1.)1045 b Ff(\003)278 2379 y Fu(T)l(o)16 b(\014nish)h(the)f(pro)q(of)h(of)g(the)f(theorem)e (w)o(e)i(pro)o(v)o(e)g(the)g(follo)o(wing)g(lemm)o(as.)228 2451 y Fn(Lemm)o(a)d(5.4.)k Fq(L)n(et)d Ft(G)h Fq(b)n(e)g(a)f(\014nite) i(solvable)h(gr)n(oup.)j(Then)15 b(ther)n(e)g(is)f(a)h(se)n(quenc)n(e) 703 2537 y Ft(e)10 b(/)i(:)c(:)g(:)i(/)i(G)914 2544 y Fo(2)945 2537 y Ft(/)f(G)1018 2544 y Fo(1)1050 2537 y Ft(/)g(G)1123 2544 y Fo(0)1157 2537 y Fu(=)j Ft(G)p eop %%Page: 20 20 20 19 bop 228 55 a Fl(20)545 b(BODO)13 b(P)m(AREIGIS)228 154 y Fq(with)18 b Ft(G)372 161 y Fi(i)386 154 y Ft(=G)448 161 y Fi(i)p Fo(+1)522 140 y Fr(\030)522 156 y Fu(=)575 154 y(\()p Fn(Z)p Ft(=p)676 161 y Fi(i)691 154 y Fn(Z)p Fu(\))744 136 y Fi(e)760 141 y Fd(i)776 154 y Fq(,)f Ft(p)832 161 y Fi(i)864 154 y Fq(prime)g(and)h Ft(G)1137 161 y Fi(i)1162 154 y Ft(/)12 b(G)18 b Fq(normal)f(sub)n(gr)n(oups.)228 237 y(Pr)n(o)n(of.)i Fu(b)o(y)c(induction.)21 b(W)l(e)15 b(only)h(indicate)f(ho)o(w)h(to)g(construct)g Ft(G)1466 244 y Fi(i)p Fo(+1)1541 237 y Fu(from)f Ft(G)1694 244 y Fi(i)1709 237 y Fu(.)228 295 y(Let)k Ft(M)f(/)13 b(G)458 302 y Fi(i)492 295 y Fu(b)q(e)19 b(a)g(normal)f(subgroup)i(of)f(prime)e (index)h Ft(p)1343 302 y Fi(i)1358 295 y Fu(.)29 b(De\014ne)19 b Ft(G)1593 302 y Fi(i)p Fo(+1)1671 295 y Fu(:=)228 316 y Fh(T)269 368 y Fi(g)q Fp(2)p Fi(G)349 353 y Ft(g)r(M)5 b(g)451 335 y Fp(\000)p Fo(1)499 353 y Fu(.)21 b(It)16 b(has)h(all)e(the)h(required)f(prop)q(erties.)440 b Ff(\003)228 441 y Fn(Lemm)o(a)14 b(5.5.)19 b Fq(L)n(et)c Ft(K)q(=k)k Fq(b)n(e)d(a)g(normal)g(sep)n(ar)n(able)g(\014nite)h(\014eld)g (extension.)24 b(L)n(et)228 499 y Ft(f)c Fr(2)c Ft(k)r Fu([)p Ft(x)p Fu(])h Fq(b)n(e)h(sep)n(ar)n(able)g(and)g(irr)n(e)n (ducible)h(of)f(de)n(gr)n(e)n(e)f Ft(p)i Fq(a)f(prime.)24 b(Then)19 b(either)228 557 y Ft(f)k Fq(is)17 b(irr)n(e)n(ducible)h (over)f Ft(K)22 b Fq(or)17 b Ft(f)22 b Fq(c)n(ompletely)d(splits)f (into)g(line)n(ar)f(factors.)228 640 y(Pr)n(o)n(of.)i Fu(Let)13 b Ft(a)487 647 y Fo(1)506 640 y Ft(;)8 b(:)g(:)g(:)f(;)h(a) 641 647 y Fi(p)674 640 y Fu(b)q(e)k(the)h(zeros)g(of)g Ft(f)18 b Fu(in)12 b(the)h(algebraic)f(closure)g(and)i(let)e Ft(G)228 698 y Fu(b)q(e)g(the)g(automorphism)e(group)j(of)g Ft(K)t Fu(\()p Ft(a)964 705 y Fo(1)983 698 y Ft(;)8 b(:)g(:)g(:)f(;)h (a)1118 705 y Fi(p)1138 698 y Fu(\))p Ft(=k)r Fu(.)20 b Ft(G)12 b Fu(op)q(erates)h(transitiv)o(ely)228 756 y(on)j(the)g(zeros,)f(since)h Ft(f)21 b Fu(is)16 b(irreducible.)j(Let)d Ft(N)21 b Fu(b)q(e)16 b(the)g(\014xgroup)h(of)f Ft(K)t Fu(.)21 b(Since)228 814 y Ft(K)q(=k)16 b Fu(is)d(normal,)g(w)o(e)g(get) g(that)h Ft(N)d(/)5 b(G)15 b Fu(is)e(a)h(normal)e(subgroup.)22 b Ft(N)d Fu(decomp)q(oses)228 873 y Fr(f)p Ft(a)279 880 y Fo(1)298 873 y Ft(;)8 b(:)g(:)g(:)f(;)h(a)433 880 y Fi(p)453 873 y Fr(g)j Fu(in)o(to)g(orbits)g(of)h(equal)e(cardinalit)o (y)g(since)g Ft(G)i Fu(op)q(erates)g(transitiv)o(ely)228 931 y(and)17 b Ft(N)k Fu(is)16 b(normal.)21 b(So)c(either)e Ft(N)22 b Fu(op)q(erates)17 b(transitiv)o(ely)d(or)j(trivially)l(.)j (Hence)228 989 y Ft(f)h Fu(is)16 b(irreducible)f(or)h(splits)g (completely)l(.)690 b Ff(\003)p eop %%Page: 21 21 21 20 bop 465 55 a Fl(F)o(ORMS)18 b(OF)e(HOPF)h(ALGEBRAS)g(AND)g (GALOIS)f(THEOR)m(Y)199 b(21)830 154 y Fs(References)228 233 y Fv([C1])19 b Fc(L.)60 b(Childs)p Fv(:)111 b(T)m(aming)58 b(Wild)h(Extensions)j(with)e(Hopf)g(Algebras.)313 283 y(T)m(rans.)13 b(Amer.)g(Math.)h(So)q(c.)f(304,)g(p.)h(111-140.)d (1987.)228 333 y([C2])19 b Fc(L.)h(Childs)p Fv(:)29 b(On)21 b(the)f(Hopf)g(Galois)e(Theory)i(for)g(Separable)g(Field)g(Extensions.) g(to)313 382 y(app)q(ear)14 b(Comm.)d(Alg.)228 432 y([CHR])19 b Fc(S.)d(Chase,)i(D.K.)e(Harrison,)h(A.)f(Rosen)o(b)q(erg)p Fv(:)24 b(Galois)15 b(Theory)i(and)g(Galois)e(Coho-)313 482 y(mology)c(of)j(Comm)n(utativ)o(e)d(Rings.)i(Mem.)f(Am.)g(Math.)i (So)q(c.)f(52)h(\(1965\).)228 532 y([CS])20 b Fc(S.)c(Chase,)h(M.)f(Sw) o(eedler)p Fv(:)24 b(Hopf)16 b(Algebras)h(and)f(Galois)f(Theory)m(.)h (LN)g(in)g(Math.)g(97.)313 582 y(Springer)e(1966.)228 632 y([GP])19 b Fc(C.)13 b(Greither,)g(B.)g(P)o(areigis)p Fv(:)k(Hopf)12 b(Galois)g(Theory)h(for)f(Separable)h(Field)g (Extensions.)313 681 y(J.)h(Algebra)g(106,)e(p.)i(239-258,)d(1987.)228 731 y([G])28 b Fc(A.)20 b(Grothendiec)o(k)p Fv(:)31 b(T)m(ec)o(hnique) 21 b(de)g(descen)o(te)h(I.)e(Seminaire)f(Bourbaki,)i(Exp.)f(190.)313 781 y(1959/60.)228 834 y([H])30 b Fc(R.)11 b(Haggenm)q(\177)-22 b(uller)p Fv(:)651 823 y(\177)646 834 y(Ub)q(er)13 b(In)o(v)n(arian)o (ten)e(separabler)h(Galoiserw)o(eiterungen)g(k)o(omm)n(uta-)313 884 y(tiv)o(er)i(Ringe.)f(Dissertation)h(Univ)o(ersit\177)-21 b(at)14 b(M)q(\177)-22 b(unc)o(hen)15 b(1979.)228 933 y([J])40 b Fc(N.)25 b(Jacobson)p Fv(:)40 b(An)25 b(Extension)g(of)f (Galois)f(Theory)j(to)e(Non-Normal)f(and)h(Non-)313 983 y(Separable)14 b(Fields.)g(Am.)e(J.)i(Math.)f(66,)g(p.)g(1-29.)g(1944.) 228 1033 y([K)o(O])20 b Fc(M.-A.)13 b(Kn)o(us,)i(M.)f(Ojanguren)p Fv(:)19 b(Theorie)14 b(de)h(la)f(Descen)o(te)i(et)f(Alg)o(\022)-20 b(ebres)15 b(d'Azuma)o(y)o(a.)313 1083 y(LN)f(in)f(Math.)h(389,)e (Springer)j(1974.)228 1133 y([P])33 b Fc(B.)23 b(P)o(areigis)p Fv(:)37 b(Descen)o(t)25 b(Theory)f(applied)f(to)g(Galois)f(Theory)m(.)h (T)m(ec)o(hnical)g(Rep)q(ort)313 1182 y(Univ.)13 b(California,)e(San)j (Diego,)f(1986.)228 1232 y([Sm])18 b Fc(C.)11 b(Small)p Fv(:)j(The)d(Group)g(of)f(Quadratic)h(Extensions.)h(J.)f(Pure)h(Appl.)e (Alg.)g(2,)g(p.83-105,)313 1282 y(395.)j(1972.)228 1332 y([S1])19 b Fc(M.)14 b(Sw)o(eedler)p Fv(:)19 b(Hopf)13 b(Algebras.)h(Benjamin)e({)i(New)g(Y)m(ork)g(1969.)228 1382 y([S2])19 b Fc(M.)i(Sw)o(eedler)p Fv(:)33 b(Structure)22 b(of)e(Purely)i(Inseparable)f(Extensions.)g(Ann.)g(Math.)f(87,)313 1432 y(p.)14 b(401-411.)d(1968.)228 1481 y([W])19 b Fc(C.)33 b(H.)g(W)m(enninger)p Fv(:)58 b(Hopf-Galois-Theorie)31 b(einer)k(Klasse)f(rein)g(inseparabler)313 1531 y(K\177)-21 b(orp)q(ererw)o(eiterungen.)16 b(Diplom)11 b(Thesis)k(Univ)o(ersit\177) -21 b(at)14 b(M)q(\177)-22 b(unc)o(hen)15 b(1984.)278 1625 y Fb(Ma)m(thema)m(tisches)h(Institut)f(der)g(Universit)1113 1622 y(\177)1112 1625 y(at)h(M)1220 1622 y(\177)1219 1625 y(unchen,)h(Germany)278 1674 y Fm(E-mail)d(addr)n(ess)s Fv(:)k Fa(pareigis@rz.mathemat)o(ik.u)o(ni-mu)o(enche)o(n.de)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF