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end
%%EndProlog
%%BeginSetup
%%Feature: *Resolution 600dpi
TeXDict begin
%%PaperSize: A4

%%EndSetup
%%Page: 1 1
1 0 bop 324 548 a Fj(Mathematisc)m(hes)37 b(Institut)1652
b(SS)37 b(2003)324 668 y(der)g(Univ)m(ersit)972 673 y(\177)973
668 y(at)f(M)1216 673 y(\177)1213 668 y(unc)m(hen)1657
b(Blatt)36 b(4)324 789 y(Prof.)h(Dr.)g(B.)g(P)m(areigis)1637
1287 y Fi(Algebra)45 b(I)t(I)394 1515 y Fh(13.)k(Sei)32
b Fg(p)g Fh(eine)h(Primzahl.)e(Zeigen)g(Sie:)609 1719
y(\(a\))48 b Ff(j)17 b Fh(Aut\()p Fe(Z)1099 1735 y Fd(p)1135
1717 y Fc(2)1170 1719 y Fh(\))p Ff(j)28 b Fh(=)f Fg(p)p
Fh(\()p Fg(p)22 b Ff(\000)h Fh(1\))603 1881 y(\(b\))49
b(Ist)30 b Fg(')e Ff(2)g Fh(Aut\()p Fe(Z)1381 1897 y
Fd(p)1417 1878 y Fc(2)1453 1881 y Fh(\))h(ein)g(Elemen)m(t)g(der)h
(Ordn)m(ung)g Fg(p)p Fh(,)f(so)h(ist)f Fg(')g Fh(eine)g(P)m(otenz)782
2001 y(der)k(Abbildung)1658 2121 y Fe(Z)1727 2138 y Fd(p)1763
2119 y Fc(2)1826 2121 y Ff(!)28 b Fe(Z)2023 2138 y Fd(p)2059
2119 y Fc(2)2094 2121 y Fg(;)2163 2099 y Fh(\026)2171
2121 y Fg(i)g Ff(7!)p 2359 2035 328 4 v 27 w Fh(\()p
Fg(p)22 b Fh(+)g(1\))p Fg(i)568 2325 y Fh(\(Hin)m(w)m(eis:)h(Man)h(k)m
(onstruiere)g(einen)f(Isomorphism)m(us)g(zwisc)m(hen)i(Aut\()p
Fe(Z)3266 2342 y Fd(p)3302 2323 y Fc(2)3338 2325 y Fh(\))e(und)568
2445 y(der)37 b(Grupp)s(e)g Fe(Z)1169 2404 y Fb(\002)1169
2478 y Fd(p)1205 2459 y Fc(2)1277 2445 y Fh(der)g(in)m(v)m(ertierbaren)
g(Elemen)m(te)g(v)m(on)h Fe(Z)2768 2462 y Fd(p)2804 2443
y Fc(2)2839 2445 y Fh(.)f(Beac)m(h)m(te)h(Aufga-)568
2566 y(b)s(e)32 b(7\).)2623 b(\(4\))394 2769 y(14.)49
b(Sei)44 b Fg(p)i Fh(eine)f(ungerade)h(Primzahl)d(und)i
Fg(G)h Fh(eine)f(nic)m(h)m(tab)s(elsc)m(he)h(Grupp)s(e)f(der)568
2889 y(Ordn)m(ung)33 b Fg(p)1026 2853 y Fa(3)1065 2889
y Fh(,)g(die)f(ein)g(Elemen)m(t)h(der)g(Ordn)m(ung)g
Fg(p)2449 2853 y Fa(2)2521 2889 y Fh(en)m(th)2707 2893
y(\177)2707 2889 y(alt.)609 3093 y(\(a\))48 b(Man)25
b(zeige,)g(da\031)g(die)f(v)m(on)h(diesem)g(Elemen)m(t)f(erzeugte)i(Un)
m(tergrupp)s(e)g(nor-)782 3213 y(mal)31 b(ist.)603 3375
y(\(b\))49 b(Man)39 b(zeige,)e(da\031)h(es)h(ein)e(Elemen)m(t)h(der)g
(Ordn)m(ung)h Fg(p)f Fh(gibt,)f(das)h(nic)m(h)m(t)g(in)782
3495 y(dieser)33 b(Un)m(tergrupp)s(e)h(liegt.)614 3657
y(\(c\))49 b(Man)33 b(zeige,)f(da\031)g Fg(G)g Fh(zu)h(der)g(Grupp)s(e)
f Fg(G)2295 3672 y Fa(1)2366 3657 y Fh(aus)h(Aufgab)s(e)f(7)g(isomorph)
f(ist.)3437 3778 y(\(6\))394 3981 y(15.)49 b(Sei)30 b
Fg(M)10 b Fh(\(2)19 b Ff(\002)f Fh(2)p Fg(;)f Fe(F)1176
3996 y Fa(3)1221 3981 y Fh(\))31 b(der)g(Ring)e(der)i(2)18
b Ff(\002)h Fh(2-Matrizen)30 b(mit)f(Ko)s(e\016zien)m(ten)i(in)f(dem)
568 4101 y(K)644 4105 y(\177)644 4101 y(orp)s(er)i Fe(F)956
4116 y Fa(3)1001 4101 y Fh(.)609 4305 y(\(a\))48 b(Man)29
b(zeige,)f(da\031)g(es)h(in)e(diesem)h(Ring)f(eine)h(in)m(v)m
(ertierbare)h(Matrix)e Fg(M)39 b Fh(der)782 4425 y(Ordn)m(ung)33
b(8)g(gibt.)603 4587 y(\(b\))49 b(Man)22 b(zeige)g(w)m(eiter,)g(da\031)
g Ff(f)p Fh(0)p Fg(;)17 b(E)6 b(;)17 b(M)5 b(;)17 b(M)2216
4551 y Fa(2)2255 4587 y Fg(;)g(M)2403 4551 y Fa(3)2443
4587 y Fg(;)g(M)2591 4551 y Fa(4)2631 4587 y Fg(;)g(M)2779
4551 y Fa(5)2819 4587 y Fg(;)g(M)2967 4551 y Fa(6)3007
4587 y Fg(;)g(M)3155 4551 y Fa(7)3194 4587 y Ff(g)22
b Fh(ein)f(K)3488 4591 y(\177)3488 4587 y(orp)s(er)782
4708 y(ist,)32 b(w)m(ob)s(ei)h(die)f(Op)s(erationen)g(v)m(on)h
Fg(M)10 b Fh(\(2)23 b Ff(\002)f Fh(2)p Fg(;)17 b Fe(F)2585
4723 y Fa(3)2630 4708 y Fh(\))33 b(induziert)f(sind.)568
4911 y(\(Aus)h(den)g(Aufgab)s(en)g(des)g(Staatsexamens)h(2002.\))918
b(\(4\))p eop
%%Page: 2 2
2 1 bop 394 548 a Fh(16.)49 b(Sei)37 b Fg(R)h Fh(=)f
Fe(Z)p Fh([[)p Fg(T)14 b Fh(]])36 b(der)i(Ring)f(der)i(formalen)d(P)m
(otenzreihen)k(mit)c(Ko)s(e\016zien)m(ten)568 668 y(in)31
b Fe(Z)p Fh(.)609 872 y(\(a\))48 b(Sei)33 b Fg(m)28 b
Ff(\032)g Fg(R)34 b Fh(ein)e(maximales)e(Ideal)j(v)m(on)g
Fg(R)q Fh(.)g(Zeigen)f(Sie,)g(da\031)h Fg(m)22 b Ff(\\)h
Fe(Z)30 b Fh(ein)782 992 y(maximales)g(Ideal)j(v)m(on)g
Fe(Z)d Fh(ist.)603 1154 y(\(b\))49 b(Bestimmen)32 b(Sie)g(die)g
(Einheitengrupp)s(e)h Fg(R)2416 1118 y Fb(\002)2475 1154
y Fh(.)614 1316 y(\(c\))49 b(Bestimmen)32 b(Sie)g(alle)f(maximalen)f
(Ideale)i(v)m(on)h Fg(R)q Fh(.)568 1519 y(\(Aus)g(den)g(Aufgab)s(en)g
(des)g(Staatsexamens)h(2002\).)918 b(\(6\))324 1748 y(Abgab)s(e:)32
b(Dienstag,)g(13.5.2003,)f(11.00)h(Uhr,)2088 1726 y(\177)2076
1748 y(Ubungsk)2449 1752 y(\177)2449 1748 y(asten)i(im)d(1.)h(Sto)s(c)m
(k)i(des)f(Ma-)324 1868 y(thematisc)m(hen)g(Instituts.)324
1988 y(Bitte)f(geb)s(en)h(Sie)f(auf)g(Ihrer)h(L)1475
1992 y(\177)1475 1988 y(osung)g(Ihren)h(Namen)e(und)h(Ihre)2753
1967 y(\177)2741 1988 y(Ubungsgrupp)s(e)h(an.)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF