%!PS-Adobe-2.0 %%Creator: dvips(k) 5.95b Copyright 2005 Radical Eye Software %%Title: D:/Backup/vvvpublic/Vorlesungen/03SS/ALGEBRA2.dvi %%CreationDate: Mon Mar 03 18:36:16 2008 %%Pages: 40 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMR17 CMR12 CMCSC10 CMMI12 CMR9 CMBX12 CMTI12 CMR8 %%+ CMMI8 CMSY10 MSBM10 CMEX10 CMSY8 CMR6 CMMI6 EUFM10 CMSY6 MSAM10 %%+ CMMI9 MSBM7 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: "C:\Programme\MiKTeX 2.5\miktex\bin\dvips.exe" %+ D:/Backup/vvvpublic/Vorlesungen/03SS/ALGEBRA2.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2008.03.03:1836 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}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{A A 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 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 40 /braceleftBigg put dup 80 /summationtext put dup 81 /producttext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 112 /radicalbig put dup 113 /radicalBig put dup 114 /radicalbigg put dup 115 /radicalBigg put dup 122 /bracehtipdownleft put dup 123 /bracehtipdownright put dup 124 /bracehtipupleft put dup 125 /bracehtipupright put readonly def /FontBBox{-24 -2960 1454 772}readonly def 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 109 /m put dup 112 /p put readonly def /FontBBox{-26 -224 1055 741}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 78 /N put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 114 /r put dup 115 /s put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 4391C9DF440285B8FC159D0E98D4258FC57892DDF0342CA1080743A076089583 6AD6FB2DC4C13F077F17789476E48402796E685107AF60A63FB0DE0266D55CF1 8D0AD65B9342CB686E564758C96164FFA711B11C1CE8C726F3C7BB1044BBD283 9AA4675747DF61E130A55E297CA5F0182A3F12F9085AF2F503481071724077A9 387E27879A9649AD5F186F33500FAC8F7FA26634BDCE1221EC0ED0E359E5EA5E 6166526FEB90C30D30099FBDC1BC2F9B62EFEEC48345160804AA98F8D0AA54B7 A480E715426651865C8E444EDB798C7E11040AF6E5A7ED1888653C6DBF5E6169 70BCD9C063B63B561EF165BF3AF11F8E519F37C6FDA2827685739DE2C48B5ADE EE84F067D704D4511DBFA49E166D543CFD9ECD7417055D8A827F51E087CD2927 BAFC7E6CFBD70B0FE969F890A11149D3D44D422C3370495DA9951AEE7253A49F 3A9444C8CD9158D84117299F7F2332FEB0F94E6ED8BC7AA789A3219BC2F227D3 3B5BC75FB53B55D72AF4A6A7BB613FA235B11BB37D059FD87127CEF73D5B3FBF 9F91ABAD78BD9240BD9525EBA78095EA0BDB25D1A19E876F292882EAD5619D46 D20317A345D931F4FF4EAE6216C27044CBA525E3B917CEA25A04C120466C4B93 FC720E6BA832A06CCA0A3916CEF0968D49085AEBD243C41A448289A6F05CE3F5 79148DC112A3CC7E8FF810B8C1A09E05F496C0F1EBA334E42E05C376C98F5F69 C06C71BFC0A2F3AC9951CFBB143C66FB84F9C4ED27DF70869352D61BD5E11508 0797B87C726ABE0FD05DB617F7B7BA6C376A8B4E29F8C74F190AA3581955B78C 63A3058AB7D47F877D3C478D201549C40E96BDA5E2EA10B65E395A2F60F2CE33 219A371E463651DB1A34B8855DE25F39A9436A9E8FC283C8FFB6EA06CDBA3204 36A8B05635B19FB72F3BB503D37BDDD0A5E4BA746A110E7729B90AFB4C8E1B14 683F2ACC9C169ACCE3CA1DFECCF879AC454DE48CC701CB78D09D0E82BC8A22C1 B3468C832B75ADBD0D0DE1031AC80AEE2C9C470C8988314BE5BE2C18566A413A C26B3EF417C60C27610E20F1F5B11770D59C470AAAA429394F851D6A9C94C076 A62C9C97968F58BCDB3D54A04C02932F86F8A41A4957974929624EB3AD0B3576 C5172C521C6DE708350212D2289F6154AC83EF526A009A2E83B671BA8058BEDA 61D800B7D878707D11219C4F2672A6E5F878F32DCB11E971DC8AD2160310DDF0 419E260EF972E98D5555C43060809A4AF38FA1D000BF434F0DC5739B2864FB84 503233D4EC2B2F4A31DBC4E40360CC5A5FFB54E4E2EB32D3A32D17498CECF131 6779322425A3E0F4B7BF99E32C6231E612A8E2E898B757A1DA8D13E2E4787455 0DCF99D646548E2B3888E2304A0F4D8C5B4D2846FCBFE9535A90A9E471A24B13 DF940ADD6D12E929F3EE39EDB839C7D9ECE4D01AF3D15322835E168C16AB3A3E 3CAF6C859DC06707D81D5636BA059B752345D2F142F89D6140CB3EC3A45BE635 3CA9D0A419BBA6BDBAF810345DEA93AF7257BE9CAEAFCC0473C71C4536AD0CF0 6F0C81B173EBA37D625E80C340A2FC9FBDEA0A2CA6963F0B53F5E0FB0FC21ED3 4559A6BB4DBA095AEE6A1E3655B3B57E69BF08B016248500B93E79E8504949A9 6A0E3BA4E4473329DE35FC0D8A477AB2F3728A1D1BA1CBE4EE031FF50D518CDD B7C1856392D8F5B1DFBA3AD13D48E8AA098CC05B9024F765E5C5073FE942A03B D98A11E3823D4D07B22F7069143F3AB6715F2ADB781DCAC5A14188BC6062CEA1 EC38F47D4427A68F56B1FA9D3533088DEBA085D912800F2B9E28D5393B3621D3 26A85C308ACA0C3986D94DDE2F19008137106AF9091323A03F4C9A69DD2D24BB 390432A75BA7D16E1DB3EE2DF7B153B6671065FDE047E064529CD52AA587E909 661A4C7A42A6A469D2F920729FFB3D4FA552098A78013D6517ACCC4A1C343505 C9EF012CE00DC51A9130EDC3A6CAF80D179DD9BF8EBB1A99E1DC2B52821E5BEF A86EEBCBA040355FD9A128F5C0A22CC31EF250F69B9E00D49E3E9A8A26B1A4E4 AAFA48DEA9857CB54A162E4B56F69CAB416754701FEC0C9985A737FE5E989DA4 DA65EC8604612761A319CC627E25B8EE77269A8B691711FF7CDA1DB82BA5F6BF BB8D1311FA1B2AC9F13CE71BA2FE33A01D66FC7D86A81F25BA847F2393ED1CCC 63CEEC8B17B6B9B925DECD6B536D9940C94508C9447AE52F2B5C6A26E343ECF8 0D84F815F2EE1A357FC7F6791D7B00DBAE39E7F9BD87541DB72B03D0AAFB4A08 30EE373E9E80547944ABC9F2C9AC9230AAB7C2D4A53799EFD3CE74891107554E 90AF0F46E9B3CBCAEB72C885FF01ECF9E4FFD9A5B5279F9D583F3A29D2AF5741 DA1109FFD93069A7756319CFF40E3CBC9D1CB5156F8A77573161283198EC40F4 99F1F02583B220A6C610890D0A288C9220F11CD5B52AB5DB59D3111E5E7BA994 FBC0848A340DC5649F2506993D7DA62028A355D0B4CA764CBC7E14923FA50BFD 2EE7B4ECD475B38AD7FDF769B4FB33CE1E22F045E0172C3058A5A9E06B6242F9 896FC86BD6CE66422538C9EDECB6DC44C1D1BF5F0E0D1BDD7EF2FB720311C506 890F089282BF4DA92F4A250A8C201129AC57E78987D4E837AAB5CBF0FF0269B7 F18B5A3F688DDE01D67BB9BBF8706E9A8C12C384F79DA2E3E14C79E854DF0A80 881417A6E49587886595C9FC97D6C0156E08D304070D32E13EFD95DB2F491EC6 1B74BE74E7A2086283386E6C2FBFFDD07CC5E4C44C09CB424F66A10989A8016C EB7C60B68610832BBBB15BAB494398AF32C11C4EE7A0E323829A8548DCCB7087 9781B177E2D509E4EBE02C2B35DCF1AFB39A41ED117AF605B132F8A80C210D58 25DA0E85586351E0D74DF9D6AB4ED0B3E452615D190BB1653D7A461D28DE94C2 7A0CFFD225D82443E22557883E64BA76A7CEAB8E4E0DE7C55C47177A170A90D3 0E70ACD4A6D4D88CC109617B07931839596B1312837B5257622BF63ED0566515 D08338D3DC85EB0868A1C3E5212586024983AB607B539AC61609A1CA1E23D0DC 7D9573E3D3CAE5C1AD39A70DF6723104A38A46415F435B7F8B4FB94230922D7B 94EFBE71DF7770F5D6FFD17C3B47606E46F0E257F6EF0D50356A97967F40A240 039C91DA021F32F91019356906164022E0FD315B3A17E4AD3DC7BC429526B4EF D804727D16A0012942A447E44EDB801EFF1CD27A652E6FA68FFAD379C93714E2 09F1AB45328AAA9214360F82613CEC1DA21E398D02C850D1D71CFFA27C1D8069 D85F022FC676C2229DB12631FCEC0931EADD008D81FA1BD26E3363EC9025FA58 C9A49C345BDE757920AB8D17F8A2E5C73BC6F9F6549B332235A286A06703FB63 E72B05EF7A96868C998D829C4EDE1C64A65E65857C550ED294E40C89CAC5DB71 FBC1FE96C796AA85A5D66AA0CDC273CEF7607C1D2593E940D6B74832492A3FF4 A99089AD8E859BBECDE9E3E913982D877E2233B22051D4F66AC647817B2AA532 AAE1F4C9651B30F5FBF5119E7DA05FCE51FDF1868246A8B9968E5B70A4EC0838 1D8DD7E5150C8115B8FF3C1C5617F47A74986B6B616AB5B86AE8B468F47FBB41 8F2D9C805EAC8F90CB912C2C0E3EF04A500AA1F74F95F5CC897A8604D5AF239E D4354F717F4660E86119C51CFA3D7EB2A49010507741D437B4C4376343EB6557 B945494F41A7887E29A5577836F4642FAC5366F395CF050AB335820981B12F23 F08221B14C9CB2CB67658D8FD6D1F0BBC65F121BAA5FDE1FC5EA93FFE0F2D04F 7FE56CFE86700A017C025F27662841675076B317CBCB70C2CC772E7DE2DB5176 9362FE521A4CBFEC3B5333FDA63A98324C9448417915D5800C63E9059E473D94 D7DE97CD91BA97173F8B4B117B32706669F715B63B943CEB58466BB2B2C636EB 14B1393D3150C2382D27F5552E7D369DE636537FFA29DB33CD7B47E66E8821BD 458E7DDB2DC86B8C936F81275FFF1EF2070998FE9FA4D8D470F803957B081EAB FD65A02CF187C82470F3DAB87CF441D01982FFF16F4C7FB7EB857A479A74E929 0778CD4579C8373AB8D8FAB1594193C3D18075381FFB836103B60CEDC7829BA8 B0AAFF647B11B57F3FAA12AA9A43FCEF5165AC1BDDDA935B710ADFE0F9C7BCF1 7F8822DBE724233BA15921008DE9FB8B2249F8B10F4B120E8331A79F891B294B 0050F2E6 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 20 /lessequal put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 112 /radical put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 36 /subsetornotdbleql put dup 37 /supersetornotdbleql put dup 45 /notbar put dup 67 /C put dup 70 /F put dup 75 /K put dup 78 /N put dup 81 /Q put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec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%!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 6 /plusminus put dup 8 /circleplus put dup 15 /bullet put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 33 /arrowright put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 81 /Q put dup 85 /U put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 27 /sigma put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 102 /f put dup 103 /g put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 25 /germandbls put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{-36 -251 1103 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 525003F3DBE5BF07B2E83E66B7F97DDD7CE0EEB75A78BD9227BF359D002B6ADB 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cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 66 /B put dup 68 /D put dup 70 /F put dup 72 /H put dup 76 /L put dup 80 /P put dup 83 /S put dup 85 /U put dup 97 /a put dup 98 /b put dup 101 /e put dup 103 /g put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{-53 -251 1139 750}readonly def currentdict 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 75 /K put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 19 /iota put dup 23 /nu put dup 27 /sigma put dup 28 /tau put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def currentdict end currentfile eexec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Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 8 /Phi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 25 /germandbls put dup 33 /exclam put dup 34 /quotedblright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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6[94 105 39[80 22[59 59 3[33 2[58 58 6[46 46{}22 99.6264 /CMEX10 rf /Fk 165[66 7[72 77 2[72 2[77 4[61 2[72 21[28 7[77 77 36[{}10 99.6264 /MSBM10 rf /Fl 143[83 1[50 3[28 2[50 50 9[66 66 72 4[62 3[81 23[55 55 0 0 2[66 66 100 7[100 100 6[100 8[77 2[77 77 77 77 77 1[50 6[77 1[77 3[77 28 77{}32 99.6264 /CMSY10 rf /Fm 137[34 41 25 33 32 32 36 1[43 62 21 37 29 24 1[34 34 1[36 31 30 37 18[56 68 48 60 13[35 55 20 20 18[46 4[33 6[40 15[45 11[{}31 66.4176 /CMMI8 rf /Fn 194[55 3[35 1[35 35 35 35 35 35 35 35 4[55 1[27 27 40[{}13 66.4176 /CMR8 rf /Fo 128[50 4[40 47 45 65 45 52 32 40 41 45 50 50 55 80 25 45 30 30 50 45 30 45 50 45 45 50 6[60 2[97 72 72 70 55 71 75 66 75 72 87 61 75 51 38 72 75 64 66 74 70 69 72 3[75 1[30 30 4[50 50 50 50 50 50 1[30 35 30 2[40 40 14[52 10[86 57 55 60 11[{}70 99.6264 /CMTI12 rf /Fp 128[56 4[50 4[62 44 44 46 1[62 56 62 93 31 59 1[31 1[56 1[51 2[62 54 11[86 1[62 2[77 3[67 3[88 1[70 1[86 1[80 8[56 56 56 56 56 56 56 56 56 56 1[31 33[62 12[{}37 99.6264 /CMBX12 rf /Fq 128[38 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b(anszendent,)148 5044 y Ft(\(2\))42 b Fr(K)7 b Ft([)p Fr(u)p Ft(])27 b Fk($)h Fr(K)7 b Ft(\()p Fr(u)p Ft(\))p Fo(,)148 5160 y Ft(\(3\))42 b Fr(K)7 b Ft(\()p Fr(u)p Ft(\))564 5132 y Fl(\030)565 5164 y Ft(=)669 5160 y Fr(K)g Ft(\()p Fr(x)p Ft(\))35 b Fo(=)g(F)-7 b(unktionenk)1555 5164 y(\177)1556 5160 y(orp)i(er)1811 5164 y(\177)1811 5160 y(ub)g(er)35 b Fr(K)7 b Fo(,)148 5276 y Ft(\(4\))42 b Fr(u)371 5240 y Fi(\000)p Fn(1)504 5276 y Fr(=)-61 b Fl(2)28 b Fr(K)7 b Ft([)p Fr(u)p Ft(])p Fo(,)148 5392 y Ft(\(5\))42 b([)p Fr(K)7 b Ft(\()p Fr(u)p Ft(\))27 b(:)h Fr(K)7 b Ft(])28 b(=)f Fl(1)p Fo(.)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 1197 -99 a Fq(Normale)27 b(und)d(separable)j(K) 2058 -96 y(\177)2058 -99 y(orp)r(ererw)n(eiterungen)1160 b(7)0 100 y Fp(Satz)60 b(7.13.)52 b Fo(Sei)g Fr(\033)64 b Ft(:)d Fr(K)68 b Fl(\000)-57 b(!)60 b Fr(L)53 b Fo(ein)f(K)1647 104 y(\177)1648 100 y(orp)-5 b(erisomorphismus,)51 b Fr(u)h Fo(ein)g(Element)g(eines)g(Erweite-)0 217 y(rungsk)277 221 y(\177)278 217 y(orp)-5 b(ers)49 b(von)f Fr(K)57 b Fo(und)49 b Fr(v)k Fo(ein)c(Element)g(eines)f(Erweiterungsk)2673 221 y(\177)2674 217 y(orp)-5 b(ers)49 b(von)g Fr(L)p Fo(.)g(Es)h(gelte)f(eine)0 333 y(der)35 b(b)-5 b(eiden)33 b(folgenden)h(Be)-5 b(dingungen:)148 469 y Ft(\(1\))42 b Fr(u)32 b Fo(ist)h(Nul)5 b(lstel)g(le)33 b(eines)f(irr)-5 b(e)g(duziblen)32 b(Polynoms)g Fr(f)38 b Fl(2)28 b Fr(K)7 b Ft([)p Fr(x)p Ft(])34 b Fo(und)f Fr(v)j Fo(ist)d(Nul)5 b(lstel)g(le)33 b(von)f Fr(\033)t Ft(\()p Fr(f)11 b Ft(\))28 b Fl(2)315 586 y Fr(L)p Ft([)p Fr(x)p Ft(])p Fo(.)148 702 y Ft(\(2\))42 b Fr(u)27 b Ft(:)h Fr(K)42 b Fo(und)34 b Fr(v)e Ft(:)c Fr(L)35 b Fo(sind)f(tr)-5 b(anszendent.)0 838 y(Dann)44 b(gibt)i(es)f(einen)f(K)954 842 y(\177)955 838 y(orp)-5 b(erisomorphismus)44 b Fr(\034)58 b Ft(:)48 b Fr(K)7 b Ft(\()p Fr(u)p Ft(\))2297 811 y Fl(\030)2298 843 y Ft(=)2422 838 y Fr(L)p Ft(\()p Fr(v)t Ft(\))45 b Fo(mit)h Fr(\034)11 b Ft(\()p Fr(u)p Ft(\))47 b(=)g Fr(v)i Fo(und)c Ft(\()p Fr(\034)11 b Fl(j)3620 802 y Fm(L)3620 864 y(K)3736 838 y Ft(:)47 b Fr(K)0 955 y Fl(\000)-57 b(!)27 b Fr(L)p Ft(\))h(=)g Fr(\033)t Fo(.)0 1120 y Fp(De\014nition)56 b(7.15.)49 b Ft(Eine)g(K)1135 1124 y(\177)1135 1120 y(orp)s(ererw)m (eiterung)h Fr(F)67 b Ft(:)54 b Fr(K)h Ft(hei\031t)48 b Fo(Zerf)2687 1124 y(\177)2688 1120 y(al)5 b(lungsk)3029 1124 y(\177)3030 1120 y(orp)-5 b(er)46 b Ft(der)i(P)m(olynome)0 1237 y Fl(f)p Fr(f)98 1252 y Fm(i)126 1237 y Fl(j)p Fr(i)28 b Fl(2)g Fr(I)8 b Fl(g)p Ft(,)27 b(w)m(enn)i(alle)f Fr(f)933 1252 y Fm(i)989 1237 y Ft(in)g Fr(F)14 b Ft([)p Fr(x)p Ft(])28 b(in)g(Linearfaktoren)g(zerfallen)h(und)f Fr(F)2753 1241 y Ft(\177)2750 1237 y(ub)s(er)g Fr(K)35 b Ft(v)m(on)29 b(den)f(Nullstellen)0 1353 y(der)33 b Fr(f)216 1368 y Fm(i)277 1353 y Ft(erzeugt)g(wird.)0 1519 y Fp(Satz)38 b(7.17.)k Fo(Sei)35 b Fr(f)j Fl(2)29 b Fr(K)7 b Ft([)p Fr(x)p Ft(])36 b Fo(nicht)f(konstant.)f(Dann)g(existiert)i(ein)e(Zerf) 2754 1523 y(\177)2755 1519 y(al)5 b(lungsk)3096 1523 y(\177)3097 1519 y(orp)-5 b(er)34 b Fr(F)49 b Fo(von)34 b Fr(f)46 b Fo(und)0 1635 y(es)35 b(ist)g Ft([)p Fr(F)41 b Ft(:)28 b Fr(K)7 b Ft(])28 b Fl(\024)g Ft(Grad)o(\()p Fr(f)11 b Ft(\)!)35 b Fo(.)0 1801 y Fp(F)-9 b(olgerung)40 b(7.19.)i Fo(Je)34 b(zwei)h(Zerf)1313 1805 y(\177)1314 1801 y(al)5 b(lungsk)1655 1805 y(\177)1656 1801 y(orp)-5 b(er)33 b(von)h Fr(f)2189 1805 y Fo(\177)2189 1801 y(ub)-5 b(er)35 b Fr(K)42 b Fo(sind)34 b(isomorph)3161 1805 y(\177)3161 1801 y(ub)-5 b(er)35 b Fr(K)7 b Fo(.)0 1967 y Fp(De\014nition)32 b(7.20.)39 b Ft(Ein)28 b(K)1037 1971 y(\177)1037 1967 y(orp)s(er)f Fr(L)h Ft(hei\031t)g Fo(algebr)-5 b(aisch)29 b(ab)-5 b(geschlossen)p Ft(,)26 b(falls)i(jedes)h(\(nic)m(h)m(t-k)m (onstan)m(te\))0 2083 y(P)m(olynom)39 b Fr(f)48 b Fl(2)37 b Fr(L)p Ft([)p Fr(x)p Ft(])i(eine)g(Nullstelle)h(\(und)e(damit)g(alle) h(Nullstellen\))g(in)g Fr(L)f Ft(b)s(esitzt,)h(d.h.)g(falls)f(jedes)3 2203 y(\177)0 2199 y(ub)s(er)33 b Fr(L)g Ft(algebraisc)m(he)h(Elemen)m (t)g(sc)m(hon)g(in)f Fr(L)g Ft(liegt.)0 2365 y Fp(Satz)k(7.22)f Ft(\(Steinitz\))p Fp(.)42 b Fo(Je)-5 b(der)34 b(K)1290 2369 y(\177)1291 2365 y(orp)-5 b(er)34 b(b)-5 b(esitzt)34 b(einen)f(\(bis)h(auf)g(Isomorphie\))e(eindeutig)i(b)-5 b(estimmten)0 2481 y(algebr)g(aischen)33 b(A)n(bschlu\031.)703 2679 y Ft(8.)48 b Fs(Normale)39 b(und)f(sep)-7 b(arable)36 b(K)2156 2675 y(\177)2152 2679 y(orperer)-9 b(weiter)n(ungen)0 2853 y Fp(De\014nition)46 b(8.1.)f Ft(Eine)40 b(algebraisc)m(he)h(K) 1608 2857 y(\177)1608 2853 y(orp)s(ererw)m(eiterung)g Fr(F)52 b Ft(:)39 b Fr(K)47 b Ft(hei\031t)39 b Fo(normal)p Ft(,)f(w)m(enn)j(jedes)f(ir-)0 2970 y(reduzible)47 b(P)m(olynom)g Fr(p)j Fl(2)g Fr(K)7 b Ft([)p Fr(x)p Ft(],)46 b(das)g(in)g Fr(F)59 b Ft(eine)47 b(Nullstelle)g(b)s(esitzt,)g(in)f Fr(F)14 b Ft([)p Fr(x)p Ft(])46 b(in)f(Linearfaktoren)0 3086 y(zerf)154 3090 y(\177)154 3086 y(allt.)0 3252 y Fp(Satz)38 b(8.2.)j Fo(Sei)35 b Fr(F)41 b Ft(:)28 b Fr(K)42 b Fo(ein)34 b(end)5 b(liche)34 b(K)1534 3256 y(\177)1535 3252 y(orp)-5 b(er)g(erweiterung.)34 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Ft(\(Emil)c(Artin\))p Fp(.)42 b Fo(Sei)35 b Fr(F)41 b Ft(:)28 b Fr(K)42 b Fo(eine)34 b(K)1773 4770 y(\177)1774 4766 y(orp)-5 b(er)g(erweiterung.)34 b(Dann)g(sind)3027 4770 y(\177)3028 4766 y(aquivalent)148 4903 y Ft(\(1\))42 b Fr(F)48 b Fo(ist)35 b(galoissch)972 4907 y(\177)972 4903 y(ub)-5 b(er)35 b Fr(K)7 b Fo(.)148 5019 y Ft(\(2\))42 b Fr(F)48 b Fo(ist)35 b(sep)-5 b(ar)g(ab)g(el,)34 b(normal)g(und)g(end) 5 b(lich)1868 5023 y(\177)1868 5019 y(ub)-5 b(er)35 b Fr(K)7 b Fo(.)148 5135 y Ft(\(3\))42 b Fr(F)48 b Fo(ist)35 b(Zerf)738 5139 y(\177)739 5135 y(al)5 b(lungsk)1080 5139 y(\177)1081 5135 y(orp)-5 b(er)33 b(eines)i(sep)-5 b(ar)g(ablen)33 b(Polynoms)2502 5139 y(\177)2502 5135 y(ub)-5 b(er)35 b Fr(K)7 b Fo(.)0 5301 y Fp(F)-9 b(olgerung)50 b(8.5.)c Fo(Sei)d Fr(L)g Ft(:)f Fr(K)50 b Fo(eine)43 b(end)5 b(liche)41 b(sep)-5 b(ar)g(able)42 b(K)2341 5305 y(\177)2342 5301 y(orp)-5 b(er)g(erweiterung.)42 b(Dann)g(gibt)h(es)g (eine)0 5417 y(galoissche)33 b(K)527 5421 y(\177)528 5417 y(orp)-5 b(er)g(erweiterung)34 b Fr(F)42 b Ft(:)28 b Fr(K)42 b Fo(mit)34 b Fr(F)42 b Fl(\023)28 b Fr(L)g Fl(\023)g Fr(K)7 b Fo(.)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 0 -99 a Fq(8)1562 b(Algebra)26 b(I)r(I)g({)g(P)n(areigis)0 100 y Fp(De\014nition)i(8.7.)35 b Ft(Ein)25 b(K)970 104 y(\177)970 100 y(orp)s(er)e Fr(K)31 b Ft(hei\031t)25 b Fo(p)-5 b(erfekt)26 b(\(vol)5 b(lkommen\))p Ft(,)22 b(w)m(enn)j(jede)g(algebraisc)m(he)g(K)3561 104 y(\177)3561 100 y(orp)s(erer-)0 217 y(w)m(eiterung)34 b Fr(F)41 b Ft(:)28 b Fr(K)40 b Ft(separab)s(el)33 b(ist.)0 395 y Fp(Satz)38 b(8.9.)j Fo(Ein)35 b(irr)-5 b(e)g(duzibles)34 b(Polynom)g Fr(p)28 b Fl(2)g Fr(K)7 b Ft([)p Fr(x)p Ft(])36 b Fo(ist)f(genau)f(dann)g(sep)-5 b(ar)g(ab)g(el,)34 b(wenn)p Fr(p)3341 359 y Fi(0)3391 395 y Fl(6)p Ft(=)28 b(0)p Fo(.)0 573 y Fp(F)-9 b(olgerung)40 b(8.11.)i Fo(Je)-5 b(der)34 b(K)1128 577 y(\177)1129 573 y(orp)-5 b(er)35 b(der)f(Char)-5 b(akteristik)34 b(0)h(ist)g(p)-5 b(erfekt.)0 752 y Fp(F)c(olgerung)40 b(8.13.)i Fo(Je)-5 b(der)34 b(end)5 b(liche)34 b(K)1512 756 y(\177)1513 752 y(orp)-5 b(er)34 b(ist)h(p)-5 b(erfekt.)0 930 y Fp(Satz)38 b(8.15.)j Fo(Sei)35 b Fr(F)41 b Ft(:)28 b Fr(K)42 b Fo(eine)34 b(K)1251 934 y(\177)1252 930 y(orp)-5 b(er)g(erweiterung.)34 b(Dann)g(sind)2505 934 y(\177)2506 930 y(aquivalent)148 1070 y Ft(\(1\))42 b Fr(F)48 b Fo(ist)35 b(einfach)f(und)h(algebr)-5 b(aisch)1588 1074 y(\177)1588 1070 y(ub)g(er)35 b Fr(K)7 b Fo(,)35 b(d.h.)f Fr(F)42 b Ft(=)27 b Fr(K)7 b Ft(\()p Fr(u)p Ft(\))34 b Fo(und)h Fr(u)27 b Ft(:)h Fr(K)42 b Fo(algebr)-5 b(aisch.)148 1186 y Ft(\(2\))42 b Fo(Es)34 b(gibt)h(nur)g(end)5 b(lich)34 b(viele)g(Zwischenk)1822 1190 y(\177)1823 1186 y(orp)-5 b(er)33 b Fr(K)i Fl(\022)28 b Fr(L)g Fl(\022)g Fr(F)14 b Fo(.)0 1365 y Fp(Satz)25 b(8.16)h Ft(\(Satz)c(v)m(om)h(primitiv)m(en)h(Elemen)m(t\))p Fp(.)35 b Fo(Je)-5 b(de)25 b(end)5 b(liche,)23 b(sep)-5 b(ar)g(able)24 b(K)2912 1369 y(\177)2913 1365 y(orp)-5 b(er)g(erweiterung)25 b Fr(F)41 b Ft(:)28 b Fr(K)0 1481 y Fo(ist)35 b(einfach:)f Fr(F)41 b Ft(=)28 b Fr(K)7 b Ft(\()p Fr(u)p Ft(\))p Fo(.)34 b(\()p Fr(u)g Fo(hei\031t)h(dann)f(ein)h Ft(primitiv)m(es)f(Elemen)m(t)j Fo(f)2654 1485 y(\177)2654 1481 y(ur)e Fr(F)14 b Fo(.\))0 1659 y Fp(Satz)45 b(8.17)h Ft(\(F)-8 b(undamen)m(talsatz)40 b(der)g(Algebra\))p Fp(.)46 b Fo(Der)40 b(K)2199 1663 y(\177)2200 1659 y(orp)-5 b(er)41 b Fk(C)g Fo(der)g(komplexen)e(Zahlen)h(ist)i(alge-)0 1776 y(br)-5 b(aisch)34 b(ab)-5 b(geschlossen.)1453 1994 y Ft(9.)48 b Fs(Endliche)38 b(K)2157 1990 y(\177)2153 1994 y(orper)0 2168 y Fp(Satz)33 b(9.1.)38 b Fo(Je)-5 b(der)31 b(end)5 b(liche)30 b(K)1167 2172 y(\177)1168 2168 y(orp)-5 b(er)30 b(b)-5 b(esitzt)31 b Fr(p)1759 2132 y Fm(n)1837 2168 y Fo(Elemente)g(f)2291 2172 y(\177)2291 2168 y(ur)g(eine)f(Primzahl)h Fr(p)c Fl(2)h Fk(N)p Fo(.)j(Zu)g(je)-5 b(der)31 b(Zahl)0 2284 y Fr(p)49 2248 y Fm(n)134 2284 y Fo(gibt)37 b(es)h(\(bis)f(auf)h(Isomorphie\))e(genau)h(einen)g(K)1983 2288 y(\177)1984 2284 y(orp)-5 b(er)37 b Fr(GF)14 b Ft(\()p Fr(p)2484 2248 y Fm(n)2531 2284 y Ft(\))37 b Fo(mit)h Fr(p)2835 2248 y Fm(n)2920 2284 y Fo(Elementen.)f Fr(GF)14 b Ft(\()p Fr(p)3676 2248 y Fm(n)3722 2284 y Ft(\))38 b Fo(ist)0 2400 y(Zerf)175 2404 y(\177)176 2400 y(al)5 b(lungsk)517 2404 y(\177)518 2400 y(orp)-5 b(er)31 b(von)i Fr(x)1009 2364 y Fm(p)1045 2341 y Fg(n)1109 2400 y Fl(\000)19 b Fr(x)28 b Fl(2)g Fk(F)1443 2415 y Fm(p)1483 2400 y Ft([)p Fr(x)p Ft(])33 b Fo(mit)g Fk(F)1861 2415 y Fm(p)1928 2400 y Ft(=)28 b Fr(GF)14 b Ft(\()p Fr(p)p Ft(\))27 b(=)h Fk(Z)p Fr(=p)p Fk(Z)p Fo(.)33 b(\()p Fr(GF)47 b Fo(steht)33 b(f)3224 2404 y(\177)3224 2400 y(ur)g(Galois-F)-7 b(eld.\))0 2579 y Fp(Satz)36 b(9.3.)41 b Fo(Seien)34 b Fr(K)g Fl(\022)29 b Fr(L)34 b Fo(end)5 b(liche)32 b(K)1504 2583 y(\177)1505 2579 y(orp)-5 b(er.)34 b(Dann)f(ist)h Fr(L)28 b Ft(:)g Fr(K)41 b Fo(galoissch,)32 b Fl(j)p Fr(L)p Fl(j)c Ft(=)f Fr(p)3206 2543 y Fm(n)3253 2579 y Fo(,)34 b Fl(j)p Fr(K)7 b Fl(j)27 b Ft(=)h Fr(p)3643 2543 y Fm(m)3743 2579 y Fo(und)0 2695 y Fr(m=n)p Fo(.)42 b(Weiter)f(ist)h Ft(Aut\()p Fr(L=K)7 b Ft(\))40 b(=)g Fl(f)p Ft(1)p Fr(;)17 b Ft(\010)1554 2659 y Fm(m)1620 2695 y Fr(;)g Ft(\(\010)1772 2659 y Fm(m)1839 2695 y Ft(\))1877 2659 y Fn(2)1916 2695 y Fr(;)g(:)g(:)g(:)f (;)h Ft(\(\010)2243 2659 y Fm(m)2309 2695 y Ft(\))2347 2659 y Fm(n)p Fi(\000)p Fn(1)2485 2695 y Fl(g)p Fo(,)41 b(wob)-5 b(ei)40 b Ft(\010)g(:)g Fr(L)g Fl(\000)-57 b(!)40 b Fr(L)p Fo(,)h Ft(\010\()p Fr(a)p Ft(\))g(=)e Fr(a)3860 2659 y Fm(p)0 2811 y Fo(der)c(F)-7 b(r)i(ob)g(enius-Homomorphismus)32 b(ist.)0 2990 y Fp(F)-9 b(olgerung)54 b(9.4.)48 b Fo(Sei)e Fr(L)j Ft(=)g Fr(GF)14 b Ft(\()p Fr(p)1414 2953 y Fm(n)1460 2990 y Ft(\))p Fo(.)46 b(Dann)f(gibt)i(es)f(zu)g(je)-5 b(dem)45 b Fr(m)i Fo(mit)f Fr(m=n)h Fo(genau)f(einen)f(Un-)0 3106 y(terk)162 3110 y(\177)163 3106 y(orp)-5 b(er)35 b Fr(K)g Fl(\022)28 b Fr(L)35 b Fo(mit)g Fl(j)p Fr(K)7 b Fl(j)27 b Ft(=)g Fr(p)1246 3070 y Fm(m)1313 3106 y Fo(.)35 b(Das)f(sind)g(genau)h(al)5 b(le)34 b(Unterk)2540 3110 y(\177)2541 3106 y(orp)-5 b(er)35 b(von)f Fr(L)p Fo(.)0 3284 y Fp(Satz)k(9.5.)43 b Fo(Sei)35 b Fr(K)42 b Fo(ein)35 b(K)994 3288 y(\177)995 3284 y(orp)-5 b(er)36 b(b)-5 b(eliebiger)34 b(Char)-5 b(akteristik.)35 b(Sei)g Fr(G)g Fo(eine)g(end)5 b(liche)34 b(Unter)-5 b(grupp)g(e)36 b(der)0 3400 y(multiplikativen)e(Grupp)-5 b(e)35 b Fr(K)29 b Fl(n)22 b(f)p Ft(0)p Fl(g)p Fo(.)35 b(Dann)e(ist)i Fr(G)g Fo(zyklisch.)1687 3840 y Fu(Algebra)i(I)s(I)526 4058 y Ft(10.)48 b Fs(Symmetrische)37 b(Funktionen)i(und)f(allgemeine)f (Pol)-7 b(ynome)0 4232 y Fp(Lemma)44 b(10.1.)g Fo(Sei)38 b Fr(K)7 b Ft(\()p Fr(x)1027 4247 y Fn(1)1067 4232 y Fr(;)17 b(x)1166 4247 y Fn(2)1206 4232 y Fr(;)g(:)g(:)g(:)f(;)h(x)1480 4247 y Fm(n)1527 4232 y Ft(\))35 b(=)f Fr(K)7 b Ft(\()p Fr(x)1893 4247 y Fn(1)1933 4232 y Ft(\)\()p Fr(x)2064 4247 y Fn(2)2104 4232 y Ft(\))17 b Fr(:)g(:)g(:)f Ft(\()p Fr(x)2383 4247 y Fm(n)2430 4232 y Ft(\))39 b Fo(der)f(K)2755 4236 y(\177)2756 4232 y(orp)-5 b(er)39 b(der)f(r)-5 b(ationalen)38 b(F)-7 b(unk-)0 4348 y(tionen)41 b(in)h Fr(n)g Fo(V)-7 b(ariablen)41 b Fr(x)1037 4363 y Fn(1)1076 4348 y Fr(;)17 b(:)g(:)g(:)f(;)h(x)1350 4363 y Fm(n)1397 4348 y Fo(.)42 b(Die)f(symmetrische)g(Grupp)-5 b(e)42 b Fr(S)2685 4363 y Fm(n)2774 4348 y Fo(op)-5 b(eriert)41 b(auf)h Fr(K)7 b Ft(\()p Fr(x)3501 4363 y Fn(1)3541 4348 y Fr(;)17 b(:)g(:)g(:)f(;)h (x)3815 4363 y Fm(n)3862 4348 y Ft(\))0 4465 y Fo(dur)-5 b(ch)35 b(K)342 4469 y(\177)343 4465 y(orp)-5 b(er)g(automorphismen)33 b(mit)1045 4687 y Fr(\033)1104 4606 y Fj(\000)1160 4620 y Fr(f)11 b Ft(\()p Fr(x)1312 4635 y Fn(1)1351 4620 y Fr(;)17 b(:)g(:)g(:)f(;)h(x)1625 4635 y Fm(n)1672 4620 y Ft(\))p 1160 4664 551 4 v 1164 4755 a Fr(g)t Ft(\()p Fr(x)1308 4770 y Fn(1)1347 4755 y Fr(;)g(:)g(:)g(:)f(;)h(x)1621 4770 y Fm(n)1668 4755 y Ft(\))1720 4606 y Fj(\001)1794 4687 y Ft(=)1907 4615 y Fr(f)11 b Ft(\()p Fr(x)2059 4631 y Fm(\033)2101 4613 y Fe(\000)p Fh(1)2184 4631 y Fn(\(1\))2279 4615 y Fr(;)17 b(:)g(:)g(:)f(;)h(x)2553 4631 y Fm(\033)2595 4613 y Fe(\000)p 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b(Die)f(P)m(olynome)0 5392 y Fr(f)38 b Fl(2)28 b Fr(K)7 b Ft([)p Fr(x)352 5407 y Fn(1)393 5392 y Fr(;)17 b(:)g(:)g(:)e(;)i(x) 666 5407 y Fm(n)714 5392 y Ft(])22 b Fl(\\)g Fr(L)33 b Ft(hei\031en)h Fo(symmetrische)g(Polynome)p Ft(.)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 1073 -99 a Fq(Symmetrisc)n(he)25 b(F)-6 b(unktionen)24 b(und)h(allgemeine)j(P)n(olynome)1035 b(9)0 100 y Fp(F)-9 b(olgerung)48 b(10.3.)e Fo(Der)41 b(K)1080 104 y(\177)1081 100 y(orp)-5 b(er)41 b(der)h(r)-5 b(ationalen)41 b(F)-7 b(unktionen)40 b(in)h Fr(n)h Fo(V)-7 b(ariablen)41 b Fr(K)7 b Ft(\()p Fr(x)3363 115 y Fn(1)3403 100 y Fr(;)17 b(x)3502 115 y Fn(2)3541 100 y Fr(;)g(:)g(:)g(:)f(;)h(x) 3815 115 y Fm(n)3862 100 y Ft(\))0 217 y Fo(ist)35 b(galoissch)546 221 y(\177)546 217 y(ub)-5 b(er)35 b(dem)f(K)1042 221 y(\177)1043 217 y(orp)-5 b(er)34 b(der)h(symmetrischen)f(r)-5 b(ationalen)34 b(F)-7 b(unktionen)33 b(in)i Fr(n)g Fo(V)-7 b(ariablen)34 b(mit)0 333 y(der)h(Galoisgrupp)-5 b(e)34 b Fr(S)813 348 y Fm(n)860 333 y 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y(Aut\()p Fr(L)p Ft(\()p Fr(\013)q Ft(\))p Fr(=L)p Ft(\))46 b Fl(\000)-60 b(!)46 b Ft(Aut\()p Fr(K)7 b Ft(\()p Fr(\013)q Ft(\))p Fr(=K)g Ft(\).)43 b(Dieser)h(ist)g(injektiv,)g(denn)h(w)m(enn)f Fr(\033)j Ft(auf)c Fr(K)7 b Ft(\()p Fr(\013)q Ft(\))43 b(die)h(Iden)m(tit)3812 2214 y(\177)3812 2210 y(at)0 2326 y(ergibt,)35 b(dann)g(auc)m(h)h(auf)e(den)i(Nullstellen)h(v)m(on)e Fr(f)45 b Ft(und)36 b(damit)f(auc)m(h)g(auf)g Fr(L)p Ft(\()p Fr(\013)q Ft(\).)g(F)-8 b(olglic)m(h)35 b(k)3475 2330 y(\177)3475 2326 y(onnen)h(wir)0 2442 y(iden)m(ti\014zieren)f (Aut\()p Fr(L)p Ft(\()p Fr(\013)q Ft(\))p Fr(=L)p Ft(\))28 b Fl(\022)g Ft(Aut\()p Fr(K)7 b Ft(\()p Fr(\013)q Ft(\))p Fr(=K)g Ft(\).)1916 b Fd(\003)0 2620 y Fp(De\014nition)44 b(14.10.)g Ft(Sei)38 b Fr(L)e Ft(:)g Fr(K)45 b Ft(eine)38 b(K)1595 2624 y(\177)1595 2620 y(orp)s(ererw)m(eiterung)h(und)f Fr(\013)e Fl(2)h Fr(L)p Ft(.)h(Der)f Fo(Exp)-5 b(onent)36 b Ft(v)m(on)i Fr(\013)g Ft(ist)0 2737 y(die)33 b(kleinste)i(nat)657 2741 y(\177)654 2737 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Fl(\023)g Fr(V)578 5432 y Fn(4)656 5417 y Fl(\023)g(f)p Fr(e)p Fl(g)p Ft(,)g(alle)h(Un)m(tergrupp)s(en)g(sind)g(normal)f(in)g (der)h(n)2700 5421 y(\177)2700 5417 y(ac)m(hst)g(gr)3046 5421 y(\177)3046 5417 y(o\031eren)f(Un)m(tergrupp)s(e)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 0 -99 a Fq(26)1524 b(Algebra)26 b(I)r(I)g({)g(P)n(areigis)0 100 y Ft(und)41 b(die)g(Restklassengrupp)s (en)j(sind)e Fr(S)1527 115 y Fn(4)1566 100 y Fr(=)-5 b(A)1683 115 y Fn(4)1763 73 y Fl(\030)1764 104 y Ft(=)1882 100 y Fk(Z)1948 115 y Fn(2)1988 100 y Ft(,)41 b Fr(A)2129 115 y Fn(4)2168 100 y Fr(=V)2274 115 y Fn(4)2355 73 y Fl(\030)2356 104 y Ft(=)2474 100 y Fk(Z)2540 115 y Fn(3)2620 100 y Ft(und)g Fr(V)2880 115 y Fn(4)2920 100 y Ft(,)f(also)h(ab)s(elsc) m(h.)h(Also)f(ist)0 217 y(Aut\()p Fr(L=K)7 b Ft(\))28 b Fl(\023)g Ft(\(Aut\()p Fr(L=K)i Fl(\\)22 b Fr(A)1209 232 y Fn(4)1249 217 y Ft(\))28 b Fl(\023)g Ft(\(Aut\()p Fr(L=K)7 b Ft(\))22 b Fl(\\)h Fr(V)2072 232 y Fn(4)2111 217 y Ft(\))28 b Fl(\023)g Ft(0)k(eine)i(Au\015)2745 221 y(\177)2745 217 y(osung)f(v)m(on)g(Aut)q(\()p Fr(L=K)7 b Ft(\).)143 b Fd(\003)0 377 y Fp(Hauptsatz)35 b(14.13)h Ft(\()869 381 y(\177)866 377 y(ub)s(er)30 b(Au\015)1269 381 y(\177)1269 377 y(osung)i(durc)m(h)f(Radik)-5 b(ale\))p Fp(.)41 b Fo(Sei)32 b Fr(F)42 b Ft(:)27 b Fr(K)40 b Fo(eine)33 b(Galoiserweiterung)f(und)0 494 y Fr(\037)p Ft(\()p Fr(K)7 b Ft(\))42 b(=)f(0)h Fo(o)-5 b(der)42 b Fr(\037)p Ft(\()p Fr(K)7 b Ft(\))41 b(=)h Fr(p)p Fo(,)g(wob)-5 b(ei)41 b Fr(p)h Fo(gr)1654 498 y(\177)1655 494 y(o\031er)h(als)e(al)5 b(le)42 b(Primteiler)g(von)g Ft([)p Fr(F)55 b Ft(:)42 b Fr(K)7 b Ft(])42 b Fo(ist.)g(Dann)g(sind)-1 614 y(\177)0 610 y(aquivalent:)148 746 y Ft(\(1\))g(Aut\()p Fr(F)8 b(=K)f Ft(\))35 b Fo(ist)g(au\015)1096 750 y(\177)1097 746 y(osb)-5 b(ar;)148 863 y Ft(\(2\))42 b Fr(F)f Ft(:)28 b Fr(K)42 b Fo(ist)35 b(dur)-5 b(ch)34 b(R)-5 b(adikale)34 b(au\015)1556 867 y(\177)1557 863 y(osb)-5 b(ar.)0 1024 y Fp(Satz)54 b(14.14)g Ft(\(v)m(on)48 b(Ab)s(el)f(\(1824\)\))p Fp(.)h Fo(Sei)g Fr(\037)p Ft(\()p Fr(K)7 b Ft(\))52 b(=)g(0)c Fo(o)-5 b(der)48 b Fr(\037)p Ft(\()p Fr(K)7 b Ft(\))52 b(=)g Fr(p)p Fo(,)c(wob)-5 b(ei)48 b Fr(p)g Fo(gr)3351 1028 y(\177)3352 1024 y(o\031er)g(als)g(al)5 b(le)0 1140 y(Primteiler)35 b(von)f Fr(n)h Fo(ist.)g(Sei)782 1297 y Fr(f)11 b Ft(\()p Fr(x)p Ft(\))28 b(=)g Fr(x)1159 1256 y Fm(n)1228 1297 y Fl(\000)23 b Fr(a)1379 1312 y Fn(1)1418 1297 y Fr(x)1473 1256 y Fm(n)p Fi(\000)p Fn(1)1633 1297 y Ft(+)f Fr(:)17 b(:)g(:)22 b Ft(+)g(\()p Fl(\000)p Ft(1\))2168 1256 y Fm(n)2215 1297 y Fr(a)2266 1312 y Fm(n)2341 1297 y Fl(2)28 b Fr(K)7 b Ft(\()p Fr(a)2614 1312 y Fn(1)2653 1297 y Fr(;)17 b(:)g(:)g(:)f(;)h(a)2923 1312 y Fm(n)2970 1297 y Ft(\)[)p Fr(x)p Ft(])0 1454 y Fo(die)39 b Ft(allgemeine)g(Gleic) m(h)m(ung)g Fr(n)p Ft(-ten)f(Grades)p Fo(.)i(Die)f Fr(a)1979 1469 y Fm(i)2047 1454 y Fo(seien)g(tr)-5 b(anszendent)2875 1458 y(\177)2875 1454 y(ub)g(er)39 b Fr(K)7 b Ft(\()p Fr(a)3271 1469 y Fn(1)3311 1454 y Fr(;)17 b(:)g(:)g(:)f(;)h(a)3581 1469 y Fm(i)p Fi(\000)p Fn(1)3699 1454 y Ft(\))40 b Fo(f)3807 1458 y(\177)3807 1454 y(ur)0 1570 y(al)5 b(le)34 b Fr(i)28 b Ft(=)g(1)p Fr(;)17 b(:)g(:)g(:)e(;)i(n)p Fo(.)35 b(Das)f(Polynom)g Fr(f)11 b Ft(\()p Fr(x)p Ft(\))35 b Fo(ist)g(genau)g(dann)f(dur)-5 b(ch)35 b(R)-5 b(adikale)34 b(au\015)3053 1574 y(\177)3054 1570 y(osb)-5 b(ar,)34 b(wenn)g Fr(n)28 b Fl(\024)g Ft(4)p Fo(.)0 1731 y(Beweis.)40 b Ft(Die)48 b(Galoisgrupp)s(e)f(v)m(on)i Fr(f)58 b Ft(ist)48 b Fr(S)1683 1746 y Fm(n)1778 1731 y Ft(nac)m(h)g(Satz)g(10.13.)f(Die)g(Grupp)s(e)h Fr(S)3175 1746 y Fm(n)3270 1731 y Ft(en)m(th)3456 1735 y(\177)3456 1731 y(alt)g Fr(A)3691 1746 y Fm(n)3786 1731 y Ft(als)0 1847 y(Normalteiler.)33 b Fr(A)666 1862 y Fm(n)744 1847 y Ft(ist)f(nac)m(h)h(Satz)f(4.29)f(genau)h(dann)g(einfac)m(h,)g(w)m (enn)i Fr(n)28 b Fl(6)p Ft(=)f(4.)32 b(Da)e Fr(A)3186 1862 y Fm(n)3265 1847 y Ft(f)3298 1851 y(\177)3295 1847 y(ur)i Fr(n)c(>)f Ft(4)32 b(nic)m(h)m(t)0 1963 y(ab)s(elsc)m(h)i(ist,)g (ist)f Fr(S)702 1978 y Fm(n)782 1963 y Ft(f)815 1967 y(\177)812 1963 y(ur)g Fr(n)c(>)f Ft(4)33 b(nic)m(h)m(t)h(au\015)1611 1967 y(\177)1611 1963 y(osbar.)f(Da)g Fr(A)2129 1978 y Fn(2)2197 1963 y Ft(=)28 b Fl(f)p Fr(e)p Fl(g)33 b Ft(und)g Fr(A)2747 1978 y Fn(3)2815 1935 y Fl(\030)2816 1967 y Ft(=)2921 1963 y Fk(Z)p Fr(=)p Ft(3)p Fk(Z)p Ft(,)h(sind)f Fr(S)3478 1978 y Fn(2)3518 1963 y Ft(,)g Fr(S)3638 1978 y Fn(3)3677 1963 y Ft(,)g(und)0 2079 y Fr(S)60 2094 y Fn(4)132 2079 y Ft(au\015)289 2083 y(\177)289 2079 y(osbar.)3279 b Fd(\003)938 2277 y Ft(15.)48 b Fs(K)m(onstr)n(uktion)39 b(regelm)2219 2273 y(\177)2216 2277 y(a\031iger)e Fr(n)p Fs(-Ecke)0 2451 y Fp(Bemerkung)29 b(15.1.)34 b Ft(Das)22 b(regelm)1318 2455 y(\177)1318 2451 y(a\031ige)i Fr(n)p Ft(-Ec)m(k)g(ist)f(genau)g(dann)g(k)m(onstruierbar,)h(w)m(enn)h(eine)e (primitiv)m(e)0 2568 y Fr(n)p Ft(-te)33 b(Einheitswurzel)i Fr(\020)g Ft(=)28 b Fr(\013)22 b Ft(+)g Fr(i\014)34 b Fl(2)28 b Fk(C)k Ft(k)m(onstruierbar)j(ist.)0 2728 y Fp(Satz)51 b(15.2.)d Fo(Das)d(r)-5 b(e)g(gelm)1018 2732 y(\177)1019 2728 y(a\031ige)44 b Fr(n)p Fo(-Eck)i(ist)g(genau)f(dann)g (konstruierb)-5 b(ar,)45 b(wenn)g(die)g(Eulersche)g Fr(')p Fo(-)0 2845 y(F)-7 b(unktion)34 b Fr(')p Ft(\()p Fr(n)p Ft(\))h Fo(eine)f(Zweierp)-5 b(otenz)33 b(ist.)0 3005 y(Beweis.)40 b Ft(Die)f(primitiv)m(e)i(Einheitswurzel)h Fr(\020)k Ft(=)38 b Fr(\013)28 b Ft(+)e Fr(i\014)45 b Fl(2)39 b Fk(C)g Ft(ist)g(genau)h(dann)f(k)m(onstruierbar)i(\()3711 3009 y(\177)3708 3005 y(ub)s(er)0 3122 y Fk(Q)p Ft(\),)i(w)m(enn)h Fr(\013)f Ft(und)g Fr(\014)48 b Ft(k)m(onstruierbar)c(sind,)g(d.h.)f (da\031)f(es)h(nac)m(h)g(Satz)g(3.6)f Fr(\013)q(;)17 b(\014)50 b Fl(2)45 b Fr(L)g Fl(\022)g Fk(R)d Ft(gibt)h(mit)0 3238 y([)p Fr(L)51 b Ft(:)f Fk(Q)p Ft(])h(=)f(2)551 3202 y Fm(n)598 3238 y Ft(.)c(Da)f Fr(i)51 b Fl(2)g Fk(Q)p Ft(\()p Fr(i)p Ft(\))46 b(mit)g([)p Fk(Q)p Ft(\()p Fr(i)p Ft(\))52 b(:)e Fk(Q)p Ft(])h(=)f(2,)c(ist)g([)p Fk(Q)p Ft(\()p Fr(\013)q(;)17 b(\014)6 b Ft(\))50 b(:)h Fk(Q)p Ft(])g(=)f(2)3165 3202 y Fm(m)3277 3242 y Ft(\177)3277 3238 y(aequiv)-5 b(alen)m(t)47 b(zu)0 3360 y([)p Fk(Q)p Ft(\()p Fr(\013)q(;)17 b(\014)6 b(;)17 b(i)p Ft(\))45 b(:)h Fk(Q)p Ft(])h(=)e([)p Fk(Q)p Ft(\()p Fr(\020)8 b(;)17 b(i)p Ft(\))46 b(:)g Fk(Q)p Ft(])g(=)g(2)1563 3324 y Fm(m)p Fn(+1)1720 3360 y Ft(.)d(T)-8 b(ats)1977 3364 y(\177)1977 3360 y(ac)m(hlic)m(h)45 b(ist)2470 3321 y Fn(1)p 2470 3337 36 4 v 2470 3395 a(2)2515 3360 y Ft(\()p Fr(\020)37 b Ft(+)p 2738 3279 51 4 v 29 w Fr(\020)7 b Ft(\))46 b(=)g Fr(\013)e Ft(mit)p 3289 3279 V 43 w Fr(\020)53 b Ft(=)46 b Fr(\020)3558 3324 y Fm(n)p Fi(\000)p Fn(1)3737 3360 y Ft(und)22 3453 y Fn(1)p 10 3469 60 4 v 10 3526 a(2)p Fm(i)79 3492 y Ft(\()p Fr(\020)29 b Fl(\000)p 289 3411 51 4 v 23 w Fr(\020)7 b Ft(\))28 b(=)f Fr(\013)q Ft(.)33 b(Das)f(ist)h(wiederum)1405 3496 y(\177)1405 3492 y(aquiv)-5 b(alen)m(t)34 b(zu)f([)p Fk(Q)p Ft(\()p Fr(\020)8 b Ft(\))27 b(:)h Fk(Q)p Ft(])g(=)g([)p Fk(Q)2657 3507 y Fm(n)2732 3492 y Ft(:)g Fl(Q)2868 3507 y Fm(n)2915 3492 y Ft(\))g(=)g(2)3134 3456 y Fm(r)3204 3492 y Ft(f)3237 3496 y(\177)3234 3492 y(ur)k(geeignetes)i Fr(r)s Ft(.)0 3609 y(Nac)m(h)f(Satz)g(12.14)f(\(2\))g(ist)h(das)1188 3613 y(\177)1188 3609 y(aquiv)-5 b(alen)m(t)34 b(zu)f Fr(')p Ft(\()p Fr(n)p Ft(\))27 b(=)h(2)2165 3573 y Fm(r)2203 3609 y Ft(.)1593 b Fd(\003)0 3770 y Fp(De\014nition)45 b(15.3.)g Ft(Ein)40 b(Primzahl)g Fr(p)f Ft(hei\031t)g Fo(F)-7 b(ermatsche)39 b(Primzahl)p Ft(,)f(w)m(enn)i Fr(p)f Ft(gesc)m(hrieb)s(en)i(w)m(erden)0 3887 y(k)-5 b(ann)33 b(als)1688 4003 y Fr(p)28 b Ft(=)g(2)1918 3962 y Fn(2)1953 3938 y Fg(m)2038 4003 y Ft(+)22 b(1)p Fr(:)0 4164 y Fp(Lemma)54 b(15.4.)49 b Fo(Sei)d Fr(p)h Fo(eine)f(Primzahl)g (der)g(F)-7 b(orm)46 b Fr(p)j Ft(=)h(2)2341 4127 y Fm(N)2439 4164 y Ft(+)31 b(1)p Fo(.)46 b(Dann)g(ist)h Fr(p)g Fo(eine)f(F)-7 b(ermatsche)0 4280 y(Primzahl.)0 4441 y(Beweis.)40 b Ft(Sei)33 b Fr(a)g Ft(ein)g(T)-8 b(eiler)34 b(v)m(on)f Fr(N)10 b Ft(.)33 b(Dann)f(ist)h(die)h(Primzahl)841 4598 y Fr(p)27 b Ft(=)h(\(2)1108 4556 y Fm(b)1142 4598 y Ft(\))1180 4556 y Fm(a)1244 4598 y Ft(+)22 b(1)27 b(=)h(\(2)1609 4556 y Fm(b)1665 4598 y Ft(+)22 b(1\)\(\(2)1975 4556 y Fm(b)2008 4598 y Ft(\))2046 4556 y Fm(a)p Fi(\000)p Fn(1)2200 4598 y Fl(\000)h Ft(\(2)2387 4556 y Fm(b)2421 4598 y Ft(\))2459 4556 y Fm(a)p Fi(\000)p Fn(2)2613 4598 y Ft(+)f Fr(:)17 b(:)g(:)22 b Ft(+)g(1\))p Fr(;)0 4755 y Ft(also)33 b Fr(a)28 b Ft(=)f(1)32 b(und)h(damit)g Fr(N)38 b Ft(=)28 b(2)1205 4718 y Fm(m)1271 4755 y Ft(.)33 b(Damit)f(ist)h Fr(p)g Ft(eine)g(F)-8 b(ermatsc)m(he)34 b(Primzahl.)846 b Fd(\003)0 4915 y Fp(Bemerkung)40 b(15.5.)i Ft(Die)32 b(einzigen)i(b)s(ek)-5 b(ann)m(ten)34 b(F)-8 b(ermatsc)m(hen)34 b(Primzahlen)g(sind)434 5080 y(3)27 b(=)h(2)663 5039 y Fn(2)698 5016 y Fh(0)759 5080 y Ft(+)22 b(1)p Fr(;)49 b Ft(5)27 b(=)h(2)1211 5039 y Fn(2)1246 5016 y Fh(1)1307 5080 y Ft(+)22 b(1)p Fr(;)49 b Ft(17)27 b(=)h(2)1808 5039 y Fn(2)1843 5016 y Fh(2)1904 5080 y Ft(+)22 b(1)p Fr(;)49 b Ft(257)27 b(=)g(2)2453 5039 y Fn(2)2488 5016 y Fh(3)2549 5080 y Ft(+)22 b(1)p Fr(;)49 b Ft(65537)27 b(=)g(2)3196 5039 y Fn(2)3231 5016 y Fh(4)3292 5080 y Ft(+)22 b(1)p Fr(:)0 5237 y Ft(Es)33 b(ist)g(sic)m(her,)i (da\031)d(die)h(Zahlen)g(2)1285 5201 y Fn(2)1320 5178 y Fg(m)1405 5237 y Ft(+)22 b(1)32 b(f)1617 5241 y(\177)1614 5237 y(ur)h Fr(m)28 b Ft(=)f(5)p Fr(;)17 b(:)g(:)g(:)f(;)h Ft(19)32 b(k)m(eine)i(Primzahlen)g(sind.)g(So)e(ist)h(et)m(w)m(a)1328 5402 y Fr(F)1391 5417 y Fn(5)1458 5402 y Ft(=)28 b(2)1611 5361 y Fn(2)1646 5338 y Fh(5)1707 5402 y Ft(+)22 b(1)27 b(=)h(641)21 b Fl(\001)h Ft(6700417)p Fr(:)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 1638 -99 a Fq(Zahlen)26 b(zur)g(Basis)h Fb(p)1561 b Fq(27)0 100 y Fp(Satz)33 b(15.6.)39 b Fo(Das)31 b(r)-5 b(e)g(gelm)977 104 y(\177)978 100 y(a\031ige)30 b Fr(n)p Fo(-Eck)h(l)1534 104 y(\177)1535 100 y(a\031t)h(sich)e(genau)h (dann)g(mit)g(Zirkel)g(und)g(Line)-5 b(al)31 b(konstruier)-5 b(en,)0 217 y(wenn)33 b Fr(n)g Fo(ein)g(Pr)-5 b(o)g(dukt)34 b(einer)f(Zweierp)-5 b(otenz)32 b(mit)i(p)-5 b(aarweise)32 b(verschie)-5 b(denen)31 b(F)-7 b(ermatschen)32 b(Primzah-)0 333 y(len)i(ist)1623 463 y Fr(n)27 b Ft(=)h(2)1861 422 y Fm(r)1899 463 y Fr(p)1948 478 y Fn(2)2004 463 y Fr(:)17 b(:)g(:)f(p)2184 478 y Fm(m)2250 463 y Fr(:)0 645 y Fo(Beweis.)40 b Fr(n)46 b Ft(w)m(erde)i(als)f(Pro)s(dukt)f(v)m(on)h(Primzahlp)s (otenzen)h(gesc)m(hrieb)s(en)h Fr(n)i Ft(=)f(2)3152 609 y Fm(i)3176 618 y Fh(1)3215 645 y Fr(p)3264 602 y Fm(i)3288 611 y Fh(2)3264 669 y Fn(2)3343 645 y Fr(:)17 b(:)g(:)f(p)3523 609 y Fm(i)3547 617 y Fg(m)3523 670 y Fm(m)3610 645 y Ft(.)46 b(Nac)m(h)0 765 y(Satz)26 b(12.4)g(ist)g(dann)g Fr(')p Ft(\()p Fr(n)p Ft(\))i(=)g Fr(')p Ft(\(2)1257 729 y Fm(i)1281 738 y Fh(1)1319 765 y Ft(\))p Fr(')p Ft(\()p Fr(p)1508 722 y Fm(i)1532 731 y Fh(2)1508 790 y Fn(2)1570 765 y Ft(\))17 b Fr(:)g(:)g(:)f(')p Ft(\()p Fr(p)1907 729 y Fm(i)1931 737 y Fg(m)1907 790 y Fm(m)1994 765 y Ft(\))26 b(eb)s(enfalls)h(ein)f(en)m(tsprec)m(hendes)k(Pro)s (dukt.)d(Jeder)0 882 y(F)-8 b(aktor)32 b(ist)h(eine)h(Zw)m(eierp)s (otenz)g(genau)e(dann,)h(w)m(enn)h(das)f(regelm)2505 886 y(\177)2505 882 y(a\031ige)g Fr(n)p Ft(-Ec)m(k)h(k)m(onstruierbar)g (ist.)0 998 y(Es)h(ist)f Fr(')p Ft(\(2)427 962 y Fm(i)455 998 y Ft(\))c(=)g(2)678 962 y Fm(i)729 998 y Fl(\001)23 b Ft(\(1)g Fl(\000)1000 959 y Fn(1)p 1000 975 36 4 v 1000 1032 a(2)1046 998 y Ft(\))34 b(immer)h(eine)f(Zw)m(eierp)s(otenz.) i(W)-8 b(eiter)35 b(ist)f Fr(')p Ft(\()p Fr(p)2853 962 y Fm(i)2881 998 y Ft(\))c(=)g Fr(p)3104 962 y Fm(i)3156 998 y Fl(\000)23 b Fr(p)3305 962 y Fm(i)p Fi(\000)p Fn(1)3458 998 y Ft(durc)m(h)35 b Fr(p)3782 962 y Fm(i)p Fi(\000)p Fn(1)0 1115 y Ft(teilbar,)j(also)g(n)m(ur)h(dann)f(eine)h(Zw)m(eierp)s (otenz,)g(w)m(enn)g Fr(i)e Ft(=)g(1)g(ist.)i(In)f(diesem)h(F)-8 b(alle)38 b(ist)g Fr(')p Ft(\()p Fr(p)p Ft(\))f(=)f Fr(p)26 b Fl(\000)g Ft(1)0 1232 y(eine)34 b(Zw)m(eierp)s(otenz)g(genau)e(dann,) h(w)m(enn)h Fr(p)28 b Ft(=)g(2)1831 1195 y Fm(N)1920 1232 y Ft(+)22 b(1.)1729 b Fd(\003)0 1414 y Fp(Bemerkung)60 b(15.7.)51 b Ft(Konstruktionen)g(des)g(regelm)2082 1418 y(\177)2082 1414 y(a\031igen)g Fr(n)p Ft(-Ec)m(ks)h(f)2775 1418 y(\177)2772 1414 y(ur)e Fr(n)58 b Ft(=)f(3)p Fr(;)17 b Ft(5)p Fr(;)g Ft(15)49 b(und)h Fr(n)58 b Ft(=)0 1530 y(2)49 1494 y Fm(m)141 1530 y Fl(\001)25 b Ft(3)p Fr(;)17 b Ft(2)336 1494 y Fm(m)428 1530 y Fl(\001)25 b Ft(5)p Fr(;)17 b Ft(2)623 1494 y Fm(m)714 1530 y Fl(\001)26 b Ft(15)37 b(sind)h(aus)g(dem)h(Altertum)f(b)s(ek)-5 b(ann)m(t.)39 b(Das)e(7-Ec)m(k)h(und)h(das)f(9-Ec)m(k)g(sind)h(nic)m(h) m(t)0 1646 y(k)m(onstruierbar.)30 b(Die)e(Konstruktion)h(des)g(5-Ec)m (ks)h(h)1918 1650 y(\177)1918 1646 y(angt)e(eng)g(mit)h(dem)g(goldenen) g(Sc)m(hnitt)g(zusammen.)0 1762 y(C.F.Gau\031)23 b(\(1777{1855\))d(hat) j(1796)f(als)h(erster)h(die)g(Konstruktion)g(des)g(regelm)2936 1766 y(\177)2936 1762 y(a\031igen)f(17-Ec)m(ks)h(angege-)0 1879 y(b)s(en.)f(Eine)h(m)499 1883 y(\177)499 1879 y(oglic)m(he)f (Konstruktion)h(des)f(regelm)1840 1883 y(\177)1840 1879 y(a\031igen)h(17-Ec)m(ks)f(geh)m(t)g(auf)g(F.J.Ric)m(helot)g (\(1808{1875\))0 1995 y(zur)138 1999 y(\177)135 1995 y(uc)m(k)31 b(\(1832\).)d(In)h(den)h(Arc)m(hiv)m(en)h(des)f(G)1587 1999 y(\177)1587 1995 y(ottinger)f(Mathematisc)m(hen)j(Seminars)e (liegt)f(ein)h(Ko\013er)f(mit)0 2111 y(einer)38 b(un)m(ter)g(Gau\031)e (en)m(tstandene)j(Dissertation)1869 2115 y(\177)1866 2111 y(ub)s(er)e(die)h(Konstruktion)g(des)g(regelm)3315 2115 y(\177)3315 2111 y(a\031igen)g(65537-)0 2227 y(Ec)m(ks.)1389 2464 y(16.)48 b Fs(Zahlen)38 b(zur)f(Basis)g Fr(p)0 2639 y Fp(Bemerkung)50 b(16.1.)d Ft(Sei)42 b Fr(p)i(>)f Ft(1)e(eine)i(nat) 1709 2643 y(\177)1706 2639 y(urlic)m(he)g(Zahl.)f(Jede)g(nat)2667 2643 y(\177)2664 2639 y(urlic)m(he)i(Zahl)d Fr(r)46 b Fl(2)e Fk(N)d Ft(l)3562 2643 y(\177)3562 2639 y(a\031t)h(sic)m(h)0 2755 y(eindeutig)34 b(in)f(der)g(Darstellung)f(zur)h(Basis)h Fr(p)669 2926 y(r)c Ft(=)e Fr(a)898 2941 y Fm(n)945 2926 y Fr(p)994 2885 y Fm(n)1063 2926 y Ft(+)22 b Fr(a)1212 2941 y Fm(n)p Fi(\000)p Fn(1)1350 2926 y Fr(p)1399 2885 y Fm(n)p Fi(\000)p Fn(1)1558 2926 y Ft(+)g Fr(:)17 b(:)g(:)k Ft(+)h Fr(a)1941 2941 y Fn(1)1981 2926 y Fr(p)2030 2885 y Fn(1)2092 2926 y Ft(+)g Fr(a)2241 2941 y Fn(0)2280 2926 y Fr(p)2329 2885 y Fn(0)2396 2926 y Ft(=)28 b(\()p Fr(a)2589 2941 y Fm(n)2636 2926 y Fr(a)2687 2941 y Fm(n)p Fi(\000)p Fn(1)2841 2926 y Fr(:)17 b(:)g(:)f(a)3023 2941 y Fn(1)3063 2926 y Fr(a)3114 2941 y Fn(0)3153 2926 y Ft(\))3191 2941 y Fm(p)0 3096 y Ft(mit)24 b(0)j Fl(\024)i Fr(a)403 3111 y Fm(i)459 3096 y Fr(<)e(p)d Ft(und)g(0)j Fr(<)h(a)1052 3111 y Fm(n)1123 3096 y Ft(darstellen.)d(Die)e(Addition)h (v)m(on)h(nat)2474 3100 y(\177)2471 3096 y(urlic)m(hen)g(Zahlen)f(in)g (der)g(Darstellung)0 3213 y(zur)33 b(Basis)g Fr(p)g Ft(erfolgt)f(wie)i (folgt:)698 3376 y Fr(r)85 b Ft(=)28 b(\()p Fr(a)1020 3391 y Fm(n)1067 3376 y Fr(a)1118 3391 y Fm(n)p Fi(\000)p Fn(1)1272 3376 y Fr(:)17 b(:)g(:)f(a)1454 3391 y Fn(1)1494 3376 y Fr(a)1545 3391 y Fn(0)1584 3376 y Ft(\))1622 3391 y Fm(p)698 3492 y Fr(s)83 b Ft(=)28 b(\()p Fr(b)1010 3507 y Fm(n)1057 3492 y Fr(b)1098 3507 y Fm(n)p Fi(\000)p Fn(1)1252 3492 y Fr(:)17 b(:)g(:)f(b)1424 3507 y Fn(1)1464 3492 y Fr(b)1505 3507 y Fn(0)1545 3492 y Ft(\))1583 3507 y Fm(p)531 3609 y Fr(r)25 b Ft(+)d Fr(s)83 b Ft(=)28 b(\()p Fr(c)1011 3624 y Fm(n)p Fn(+1)1148 3609 y Fr(c)1190 3624 y Fm(n)1253 3609 y Fr(:)17 b(:)g(:)f(c)1426 3624 y Fn(1)1466 3609 y Fr(c)1508 3624 y Fn(0)1547 3609 y Ft(\))1585 3624 y Fm(p)674 3818 y Fr(c)716 3833 y Fm(i)827 3818 y Ft(=)931 3648 y Fj(\()1011 3751 y Fr(a)1062 3766 y Fm(i)1113 3751 y Ft(+)22 b Fr(b)1252 3766 y Fm(i)1302 3751 y Ft(+)g Fr(d)1451 3766 y Fm(i)p Fi(\000)p Fn(1)1569 3751 y Fr(;)1011 3891 y(a)1062 3906 y Fm(i)1113 3891 y Ft(+)g Fr(b)1252 3906 y Fm(i)1302 3891 y Ft(+)g Fr(d)1451 3906 y Fm(i)p Fi(\000)p Fn(1)1592 3891 y Fl(\000)g Fr(p;)1891 3818 y(d)1942 3833 y Fm(i)1998 3818 y Ft(=)2101 3648 y Fj(\()2182 3751 y Ft(0)p Fr(;)130 b Ft(falls)32 b Fr(a)2642 3766 y Fm(i)2693 3751 y Ft(+)22 b Fr(b)2832 3766 y Fm(i)2883 3751 y Ft(+)g Fr(d)3032 3766 y Fm(i)p Fi(\000)p Fn(1)3177 3751 y Fr(<)28 b(p;)2182 3891 y Ft(1)p Fr(;)130 b Ft(falls)32 b Fr(a)2642 3906 y Fm(i)2693 3891 y Ft(+)22 b Fr(b)2832 3906 y Fm(i)2883 3891 y Ft(+)g Fr(d)3032 3906 y Fm(i)p Fi(\000)p Fn(1)3177 3891 y Fl(\025)29 b Fr(p;)0 4093 y Ft(Die)j(Multiplik)-5 b(ation)34 b(hat)e(eine)1191 4097 y(\177)1191 4093 y(ahnlic)m(he)i(Besc)m(hreibung)h(des)f(\\)2401 4071 y(\177)2389 4093 y(Ub)s(ertragens")e(auf)h(andere)g(Stellen.)0 4275 y Fp(Bemerkung)f(16.2)d Ft(\(Zur)d(Bestimm)m(ung)h(der)f (Zi\013ern)g(in)f(der)h(Darstellung)g(zur)g(Basis)h Fr(p)p Ft(.\))p Fp(.)36 b Ft(Wir)26 b(sc)m(hrei-)0 4392 y(b)s(en)40 b(die)h(Elemen)m(te)h(v)m(on)f Fk(Z)p Fr(=p)1149 4356 y Fm(n)1196 4392 y Fk(Z)f Ft(als)g Fl(f)p 1506 4316 49 4 v Ft(0)p Fr(;)p 1599 4316 V 17 w Ft(1)o Fr(;)17 b(:)g(:)g(:)f(;)p 1866 4337 52 4 v 17 w(a;)h(:)g(:)g(:)f(;)p 2136 4316 267 4 v 17 w(p)2185 4363 y Fm(n)2254 4392 y Fl(\000)22 b Ft(1)p 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Fk(Z)1314 5390 y Fl(\030)1315 5421 y Ft(=)1419 5417 y Fl(f)p Ft(0)p Fr(;)17 b Ft(1)p Fr(;)g(:)g(:)g(:)e(;)i(a;)g(:)g (:)g(:)e(;)i(p)2147 5376 y Fm(n)p Fi(\000)p Fn(1)2306 5417 y Fl(\000)23 b Ft(1)p Fl(g)k(\000)-59 b(!)27 b Fk(Z)p Fr(=p)2841 5376 y Fm(n)2888 5417 y Fk(Z)p Fr(:)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 0 -99 a Fq(28)1524 b(Algebra)26 b(I)r(I)g({)g(P)n(areigis)0 100 y Ft(Beac)m(h)m(te,)44 b(da\031)f(durc)m(h)g(die)h(eindeutige)g(F)-8 b(estlegung)43 b(der)g(Repr)2405 104 y(\177)2405 100 y(asen)m(tan)m(ten)i(k)m(eine)f (W)-8 b(ohlde\014niertheit)0 217 y(gepr)187 221 y(\177)184 217 y(uft)33 b(w)m(erden)h(m)m(u\031)f(und)g(da\031)f Fr(\023)1299 232 y Fm(n)p Fi(\000)p Fn(1)1470 217 y Ft(k)m(ein)i (Homomorphism)m(us)g(ist.)f(Es)h(ist)f(jedo)s(c)m(h)1014 379 y Fr(\027)1062 394 y Fm(n)p Fi(\000)p Fn(1)1200 379 y Fr(\023)1234 394 y Fm(n)p Fi(\000)p Fn(1)1399 379 y Ft(=)28 b(id)1584 396 y Fa(Z)p Fm(=p)1699 377 y Fg(n)p Fe(\000)p Fh(1)1820 396 y Fa(Z)1896 379 y Ft(:)g Fk(Z)p 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Ft(abgebildet)f(auf)g Fr(a)27 b Fl(2)i(f)p Ft(0)p Fr(;)17 b Ft(1)p Fr(;)g(:)g(:)g(:)d(;)j(p)22 b Fl(\000)h Ft(1)p Fl(g)p Ft(.)32 b(Es)h(sei)h Fr(\014)3021 1159 y Fm(n)p Fi(\000)p Fn(1)3186 1144 y Ft(=)27 b Fr(\015)3340 1159 y Fm(n)p Fi(\000)p Fn(1)3477 1144 y Fr(\013)3539 1159 y Fm(n)p Fi(\000)p Fn(1)3676 1144 y Ft(.)0 1260 y(Die)32 b(Zahl)p 631 1327 47 4 v 631 1382 a Fr(r)e Ft(=)p 809 1295 1598 4 v 28 w(\()p Fr(a)898 1397 y Fm(n)945 1382 y Fr(p)994 1353 y Fm(n)1063 1382 y Ft(+)22 b Fr(a)1212 1397 y Fm(n)p Fi(\000)p Fn(1)1350 1382 y Fr(p)1399 1353 y Fm(n)p Fi(\000)p Fn(1)1558 1382 y Ft(+)g Fr(:)17 b(:)g(:)k Ft(+)h Fr(a)1941 1397 y Fn(1)1981 1382 y Fr(p)2030 1353 y Fn(1)2092 1382 y Ft(+)g Fr(a)2241 1397 y Fn(0)2280 1382 y Fr(p)2329 1353 y Fn(0)2369 1382 y Ft(\))27 b(=)p 2538 1295 732 4 v 28 w(\()p Fr(a)2627 1397 y Fm(n)2674 1382 y Fr(a)2725 1397 y Fm(n)p Fi(\000)p Fn(1)2879 1382 y Fr(:)17 b(:)g(:)f(a)3061 1397 y Fn(1)3101 1382 y Fr(a)3152 1397 y Fn(0)3191 1382 y Ft(\))3229 1397 y Fm(p)0 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y(Wir)d(zeigen)h(jetzt,)f(da\031)g(die)h(gew)1208 1972 y(\177)1208 1968 y(ohnlic)m(he)h(Norm)e Fl(j)p Fr(x)p Fl(j)1984 1972 y Ft(\177)1984 1968 y(aquiv)-5 b(alen)m(t)26 b(ist)g(zur)f(gegeb)s(enen)h(Norm)f Fl(jj)p Fr(x)p Fl(jj)p Ft(.)f(Dazu)0 2084 y(b)s(etrac)m(h)m(ten)40 b(wir)f Fr(a)716 2099 y Fm(i)771 2084 y Fl(\000)26 b Fr(a)925 2099 y Fm(j)1000 2084 y Ft(und)39 b(un)m(tersuc)m(hen)j(die)d(Cauc)m(h)m(yfolgen)m(b)s (edingung.)j(Da)37 b Fr(\013)i(>)e Ft(0)i(und)g(damit)0 2200 y(die)33 b(F)-8 b(unktion)33 b Fk(R)645 2215 y Fn(+)732 2200 y Fl(3)28 b Fr(x)g Fl(7!)f Fr(x)1091 2164 y Fm(\013)1169 2200 y Fl(2)h Fk(R)1335 2215 y Fn(+)1426 2200 y Ft(streng)34 b(monoton)e(w)m(ac)m(hsend)j(ist,)e(ist)1219 2359 y Fl(j)p Fr(a)1298 2374 y Fm(i)1348 2359 y Fl(\000)23 b Fr(a)1499 2374 y Fm(j)1536 2359 y Fl(j)k Fr(<)h(")55 b Fl(\()-17 b(\))55 b(j)p Fr(a)2113 2374 y Fm(i)2163 2359 y Fl(\000)22 b Fr(a)2313 2374 y Fm(j)2350 2359 y Fl(j)2378 2318 y Fm(\013)2455 2359 y Fr(<)28 b(")2605 2318 y Fm(\013)2654 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v 246 1880 a Fg(k)241 1855 y Fi(p)p 300 1855 33 3 v 48 x Fm(")370 1864 y Fr(<)g(p)p Ft(,)j(w)m(ob)s(ei)g Fr(p)901 1828 y Fm(k)944 1864 y Ft(die)g(gr)1185 1868 y(\177)1185 1864 y(o\031te)f Fr(p)g Ft(P)m(otenz)i(ist,)f(die)g Fr(a)2157 1879 y Fm(i)2201 1864 y Fl(\000)16 b Fr(a)2345 1879 y Fm(j)2412 1864 y Ft(teilt.)30 b(Beac)m(h)m(te:)h(f)3066 1868 y(\177)3063 1864 y(ur)f(rationale)f Fr(a)3639 1879 y Fm(i)3697 1864 y Ft(k)-5 b(ann)0 1996 y Fr(k)36 b Ft(auc)m(h)d (negativ)g(sein:)1465 2167 y Fr(a)27 b Ft(=)1657 2100 y Fr(r)p 1657 2144 47 4 v 1657 2235 a(s)1741 2167 y Ft(=)1855 2100 y Fr(p)1904 2063 y Fm(u)1949 2100 y Fr(r)1996 2063 y Fi(0)p 1855 2144 164 4 v 1857 2235 a Fr(p)1906 2206 y Fm(v)1947 2235 y Fr(s)1993 2206 y Fi(0)2056 2167 y Ft(=)h Fr(p)2209 2126 y Fm(u)p Fi(\000)p Fm(v)2355 2100 y Fr(r)2402 2063 y Fi(0)p 2355 2144 70 4 v 2355 2235 a Fr(s)2401 2206 y Fi(0)0 2361 y Ft(mit)34 b(\()p Fr(p;)17 b(r)358 2325 y Fi(0)381 2361 y Ft(\))29 b(=)h(\()p Fr(p;)17 b(s)731 2325 y Fi(0)754 2361 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1699 y Fm(p)2905 1684 y Fr(;)17 b Fl(j)p Fr(b)3018 1643 y Fi(0)3018 1708 y Fm(i)3068 1684 y Fl(\000)23 b Fr(b)3209 1699 y Fm(i)3237 1684 y Fl(j)3265 1699 y Fm(p)3305 1684 y Ft(\))k Fr(<)h(")0 1856 y Ft(f)33 1860 y(\177)30 1856 y(ur)k Fr(i)c(>)g Ft(max)q(\()p Fr(n)597 1871 y Fm(a)638 1856 y Fr(;)17 b(n)740 1871 y Fm(b)775 1856 y Ft(\).)0 1984 y(Seien)37 b Fr(a;)17 b(b)33 b Fl(2)g Fk(Q)603 1999 y Fm(p)679 1984 y Ft(mit)j Fr(a)d Ft(=)p 1054 1897 156 4 v 33 w(\()p Fr(a)1143 1999 y Fm(i)1171 1984 y Ft(\))i(und)h Fr(b)e Ft(=)p 1625 1897 146 4 v 32 w(\()p Fr(b)1704 1999 y Fm(i)1733 1984 y Ft(\).)i(Wir)f (de\014nieren)j Fr(a)24 b Fl(\001)g Fr(b)36 b Ft(als)g(die)3008 1962 y(\177)2996 1984 y(Aquiv)-5 b(alenzklasse,)39 b(die)0 2100 y(v)m(on)33 b(\()p Fr(a)273 2115 y Fm(i)324 2100 y Fl(\001)22 b Fr(b)415 2115 y Fm(i)443 2100 y Ft(\))33 b(repr)687 2104 y(\177)687 2100 y(asen)m(tiert)h(wird.)f(Das)f(ist)i (eine)f(Cauc)m(h)m(yfolge,)i(denn)152 2272 y Fl(j)p Fr(a)231 2287 y Fm(i)281 2272 y Fl(\001)22 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Fm(k)r(i)3054 1406 y Fl(j)3082 1421 y Fm(p)3162 1406 y Fr(<)39 b("=)p Ft(2.)g(Damit)g(ist)0 1523 y Fl(j)p Fr(a)79 1538 y Fm(j)t(i)162 1523 y Fl(\000)22 b Fr(a)312 1538 y Fm(k)r(i)379 1523 y Fl(j)27 b Fr(<)h(")k Ft(f)649 1527 y(\177)646 1523 y(ur)h(gen)920 1527 y(\177)917 1523 y(ugend)g(gro\031e)f Fr(i;)17 b(j;)g(k)s Ft(.)33 b(Es)h(folgt)d(f)2134 1527 y(\177)2131 1523 y(ur)i Fr(j)h(>)27 b(l)445 1699 y Fl(j)p Fr(a)524 1714 y Fm(j)t(n)600 1724 y Fg(j)658 1699 y Fl(\000)c Fr(a)809 1714 y Fm(l)q(n)874 1726 y Fg(l)902 1699 y Fl(j)930 1714 y Fm(p)997 1699 y Ft(=)k Fl(j)p Fr(a)1179 1714 y Fm(j)t(n)1255 1724 y Fg(j)1313 1699 y Fl(\000)c Fr(a)1464 1714 y Fm(l)q(n)1529 1724 y Fg(j)1587 1699 y Ft(+)g Fr(a)1737 1714 y Fm(l)q(n)1802 1724 y Fg(j)1860 1699 y Fl(\000)g Fr(a)2011 1714 y Fm(l)q(n)2076 1726 y Fg(l)2104 1699 y Fl(j)2132 1714 y Fm(p)2199 1699 y Fl(\024)28 b Ft(max)q(\()p Fl(j)p Fr(a)2603 1714 y Fm(j)t(n)2679 1724 y Fg(j)2737 1699 y Fl(\000)22 b Fr(a)2887 1714 y Fm(l)q(n)2952 1724 y Fg(j)2989 1699 y Fl(j)3017 1714 y Fm(p)3056 1699 y Fr(;)17 b(p)3149 1658 y Fi(\000)p Fm(j)3240 1699 y Ft(\))28 b Fr(<)f(")0 1892 y Ft(f)33 1896 y(\177)30 1892 y(ur)33 b(gen)304 1896 y(\177)301 1892 y(ugend)h(gro\031e)f Fr(j)39 b Ft(und)34 b Fr(l)h Ft(und)f(damit)f Fr(n)1725 1907 y Fm(j)1795 1892 y Ft(und)g Fr(n)2048 1907 y Fm(l)2075 1892 y Ft(.)g(W)-8 b(eiter)34 b(ist)p 2586 1806 239 4 v 33 w(\()p Fr(a)2675 1907 y Fm(j)t(n)2751 1917 y Fg(j)2787 1892 y Ft(\))f(der)h(Limes)g(der)g(F)-8 b(olge)32 b(\()p Fr(a)3825 1907 y Fm(j)3862 1892 y Ft(\))0 2016 y(w)m(egen)i Fl(j)p Fr(a)369 2031 y Fm(j)t(i)452 2016 y Fl(\000)22 b Fr(a)602 2031 y Fm(j)t(n)678 2041 y Fg(j)714 2016 y Fl(j)742 2031 y Fm(p)809 2016 y Fr(<)28 b(p)962 1979 y Fi(\000)p Fm(j)1086 2016 y Ft(f)1119 2020 y(\177)1116 2016 y(ur)k(alle)h Fr(j)38 b Ft(und)c(gen)1842 2020 y(\177)1839 2016 y(ugend)f(gro\031e)f Fr(i)p Ft(.)1377 b Fd(\003)0 2198 y Fp(De\014nition)38 b(19.4.)k Ft(Sei)33 b Fr(')1016 2213 y Fm(p)1088 2198 y Ft(die)g Fr(p)p Ft(-adisc)m(he)h (Bew)m(ertung)g(auf)f Fk(Q)p Ft(.)g(Der)f(K)2722 2202 y(\177)2722 2198 y(orp)s(er)1667 2370 y Fk(Q)1744 2385 y Fm(p)1812 2370 y Ft(:=)27 b(\012)2012 2385 y Fm(p)2052 2370 y Fr(=)h Fl(\030)p Fr(;)0 2542 y Ft(hei\031t)33 b Fo(K)318 2546 y(\177)319 2542 y(orp)-5 b(er)34 b(der)h Fr(p)p Fo(-adischen)e(Zahlen)p Ft(.)0 2725 y Fp(Lemma)39 b(19.5.)j Fo(Die)34 b(Menge)1430 2856 y Fk(Z)1496 2871 y Fm(p)1564 2856 y Ft(:=)1694 2775 y Fj(\010)1752 2856 y Fr(x)28 b Fl(2)h Fk(Q)2007 2871 y Fm(p)2047 2771 y Fj(\014)2047 2831 y(\014)2080 2856 y Fl(j)p Fr(x)p Fl(j)2191 2871 y Fm(p)2258 2856 y Fl(\024)f Ft(1)2412 2775 y Fj(\011)0 3007 y Fo(ist)35 b(ein)f(Inte)-5 b(grit)613 3011 y(\177)614 3007 y(atsring)35 b(und)f(hei\031t)i Ft(Ring)c(der)h Fr(p)p Ft(-adisc)m(hen)h(ganzen)f(Zahlen)p Fo(.)0 3190 y(Beweis.)40 b Ft(W)-8 b(enn)23 b Fr(x;)17 b(y)31 b Fl(2)d Fk(Z)968 3205 y Fm(p)1008 3190 y Ft(,)22 b(dann)g(ist)g Fr(')1479 3205 y Fm(p)1518 3190 y Ft(\()p Fr(xy)t Ft(\))28 b(=)f Fr(')1896 3205 y Fm(p)1936 3190 y Ft(\()p Fr(x)p Ft(\))p Fr(')2131 3205 y Fm(p)2171 3190 y Ft(\()p Fr(y)t Ft(\))f Fl(\024)i Ft(1)22 b(und)g Fr(')2749 3205 y Fm(p)2789 3190 y Ft(\()p Fr(x)p Ft(+)p Fr(y)t Ft(\))27 b Fl(\024)h Ft(max\()p Fr(')3463 3205 y Fm(p)3503 3190 y Ft(\()p Fr(x)p Ft(\))p Fr(;)17 b(')3742 3205 y Fm(p)3781 3190 y Ft(\()p Fr(y)t Ft(\)\))27 b Fl(\024)0 3306 y Ft(1.)32 b(Damit)h(ist)g Fk(Z)613 3321 y Fm(p)680 3306 y Fl(\022)28 b Fk(Q)862 3321 y Fm(p)935 3306 y Ft(ein)33 b(Un)m(terring.)2293 b Fd(\003)0 3489 y Fp(Lemma)43 b(19.6.)g Fo(Sei)37 b Fr(a)d Fl(2)f Fk(Q)38 b Fo(mit)g Fl(j)p Fr(a)p Fl(j)1427 3504 y Fm(p)1499 3489 y Fl(\024)33 b Ft(1)p Fo(.)38 b(Dann)f(gibt)g(es) h(zu)g(je)-5 b(dem)36 b Fr(i)i Fo(eine)f(ganze)g(Zahl)g Fr(r)f Fl(2)d Fk(Z)p Fo(,)38 b(so)0 3605 y(da\031)d Fl(j)p Fr(r)24 b Fl(\000)f Fr(a)p Fl(j)462 3620 y Fm(p)529 3605 y Fl(\024)28 b Fr(p)683 3569 y Fi(\000)p Fm(i)766 3605 y Fo(.)35 b(Die)f(Zahl)g Fr(r)k Fo(kann)c(aus)h(der)g(Menge)f Fl(f)p Ft(0)p Fr(;)17 b Ft(1)p Fr(;)g Ft(2)p Fr(;)g(:)g(:)g(:)d(;)j(p) 2764 3569 y Fm(i)2814 3605 y Fl(\000)23 b Ft(1)p Fl(g)34 b Fo(gew)3201 3609 y(\177)3202 3605 y(ahlt)h(wer)-5 b(den.)0 3793 y(Beweis.)40 b Ft(Sei)g Fr(a)e Ft(=)742 3754 y Fm(b)p 742 3770 31 4 v 742 3828 a(c)821 3793 y Ft(mit)h(\()p Fr(b;)17 b(c)p Ft(\))39 b(=)f(1)h(gek)1596 3797 y(\177)1593 3793 y(urzt.)h(Da)e Fl(j)p Fr(a)p Fl(j)2102 3808 y Fm(p)2180 3793 y Fl(\024)h Ft(1,)g(ist)g Fr(c)g Ft(nic)m(h)m(t)h(durc)m(h)g Fr(p)f Ft(teilbar,)g(also)g(ist)0 3916 y(\()p Fr(p)87 3880 y Fm(i)115 3916 y Fr(;)17 b(c)p Ft(\))27 b(=)h(1.)k(Es)i(gibt)e Fr(m;)17 b(n)28 b Fl(2)g Fk(Z)33 b Ft(mit)g Fr(mc)22 b Ft(+)g Fr(np)1757 3880 y Fm(i)1813 3916 y Ft(=)28 b(1.)0 4032 y(Wir)d(b)s(etrac)m(h)m(ten)h Fr(m)f Ft(als)h(Appro)m(ximation)g (des)f(In)m(v)m(ersen)j(v)m(on)d Fr(c)p Ft(,)g(denn)h Fr(mc)i Fl(\021)g Ft(1)66 b(mo)s(d)34 b Fr(p)3268 3996 y Fm(i)3320 4032 y Ft(und)26 b Fr(p)3557 3996 y Fm(i)3585 4032 y Fr(=mc)6 b Fl(\000)g Ft(1)0 4148 y(impliziert)34 b Fl(j)p Fr(mc)22 b Fl(\000)h Ft(1)p Fl(j)793 4163 y Fm(p)860 4148 y Fl(\024)28 b Fr(p)1014 4112 y Fi(\000)p Fm(i)1097 4148 y Ft(.)0 4270 y(Sei)33 b Fr(r)e Ft(:=)c Fr(bm)p Ft(.)34 b(Dann)e(sollte)h Fr(r)i Ft(eine)f(gute)e(Appro)m (ximation)i(v)m(on)2434 4231 y Fm(b)p 2433 4247 V 2433 4304 a(c)2506 4270 y Ft(sein.)g(T)-8 b(ats)2916 4274 y(\177)2916 4270 y(ac)m(hlic)m(h)35 b(ist)386 4505 y Fl(j)p Fr(r)24 b Fl(\000)f Fr(a)p Fl(j)661 4520 y Fm(p)728 4505 y Ft(=)28 b Fl(j)p Fr(bm)22 b Fl(\000)1118 4437 y Fr(b)p 1118 4482 42 4 v 1118 4573 a(c)1170 4505 y Fl(j)1198 4520 y Fm(p)1265 4505 y Ft(=)28 b Fl(j)1407 4437 y Fr(b)p 1407 4482 V 1407 4573 a(c)1458 4505 y Fl(j)1486 4520 y Fm(p)1525 4505 y Fl(j)p Fr(mc)23 b Fl(\000)f Ft(1)p Fl(j)1879 4520 y Fm(p)1946 4505 y Fl(\024)28 b(j)p Fr(mc)22 b Fl(\000)h Ft(1)p Fl(j)2405 4520 y Fm(p)2472 4505 y Ft(=)k Fl(j)p Fr(np)2710 4464 y Fm(i)2738 4505 y Fl(j)2766 4520 y Fm(p)2833 4505 y Ft(=)h Fl(j)p Fr(n)p Fl(j)3051 4520 y Fm(p)3090 4505 y Fr(p)3139 4464 y Fi(\000)p Fm(i)3250 4505 y Fl(\024)g Fr(p)3404 4464 y Fi(\000)p Fm(i)3487 4505 y Fr(:)0 4717 y Ft(Sc)m(hlie\031lic)m(h)39 b(k)532 4721 y(\177)532 4717 y(onnen)f(wir)f(Vielfac)m(he)h(v)m(on)f Fr(p)1657 4681 y Fm(i)1722 4717 y Ft(zu)g Fr(r)i Ft(addieren,)f(um)f (eine)g(ganze)h(Zahl)e(zwisc)m(hen)j(0)d(und)0 4833 y Fr(p)49 4797 y Fm(i)99 4833 y Fl(\000)23 b Ft(1)32 b(zu)h(erhalten,)h (die)f(w)m(eiterhin)h Fl(j)p Fr(r)25 b Fl(\000)d Fr(a)p Fl(j)1674 4848 y Fm(p)1742 4833 y Fl(\024)28 b Fr(p)1896 4797 y Fi(\000)p Fm(i)2011 4833 y Ft(erf)2125 4837 y(\177)2122 4833 y(ullt.)1528 b Fd(\003)0 5016 y Fp(Satz)38 b(19.7.)k Fo(Je)-5 b(des)34 b(Element)h Fr(a)g Fo(in)g Fk(Z)1435 5031 y Fm(p)1510 5016 y Fo(b)-5 b(esitzt)36 b(genau)e(eine)h (Cauchyfolge)f Ft(\()p Fr(a)2928 5031 y Fm(i)2956 5016 y Fl(j)p Fr(i)28 b Fl(2)h Fk(N)3212 5031 y Fn(0)3251 5016 y Ft(\))35 b Fo(als)g(R)-5 b(epr)3675 5020 y(\177)3676 5016 y(asen-)0 5132 y(tanten,)35 b(f)364 5136 y(\177)364 5132 y(ur)g(die)f(gilt:)148 5276 y Ft(\(1\))42 b(0)27 b Fl(\024)h Fr(a)547 5291 y Fm(i)603 5276 y Fr(<)g(p)756 5240 y Fm(i)p Fn(+1)909 5276 y Fo(f)939 5280 y(\177)939 5276 y(ur)35 b Fr(i)28 b Ft(=)g(0)p Fr(;)17 b Ft(1)p Fr(;)g Ft(2)p Fr(;)g Ft(3)p Fr(;)g(:)g(:)g(:)m Fo(;)148 5392 y Ft(\(2\))42 b Fr(a)366 5407 y Fm(i)422 5392 y Fl(\021)28 b Fr(a)578 5407 y Fm(i)p Fn(+1)763 5392 y Ft(mo)s(d)33 b Fr(p)1032 5356 y Fm(i)p Fn(+1)1185 5392 y Fo(f)1215 5396 y(\177)1215 5392 y(ur)i Fr(i)28 b Ft(=)g(0)p Fr(;)17 b Ft(1)p Fr(;)g Ft(2)p Fr(;)g Ft(3)p Fr(;)g(:)g(:)g(:)m Fo(.)p eop end %%Page: 38 38 TeXDict begin 38 37 bop 0 -99 a Fq(38)1524 b(Algebra)26 b(I)r(I)g({)g(P)n(areigis)0 100 y Fo(Beweis.)40 b Ft(Eindeutigk)m(eit:) 33 b(Sei)e(\()p Fr(b)1226 115 y Fm(i)1255 100 y Ft(\))f(eine)h(w)m (eitere)h(F)-8 b(olge,)30 b(die)h(\(1\))g(und)f(\(2\))g(erf)2905 104 y(\177)2902 100 y(ullt.)h(Sei)g Fr(a)3312 115 y Fm(i)3336 124 y Fh(0)3403 100 y Fl(6)p Ft(=)d Fr(b)3548 115 y Fm(i)3572 124 y Fh(0)3611 100 y Ft(.)i(Dann)0 217 y(ist)36 b(auc)m(h)g Fr(a)423 232 y Fm(i)447 241 y Fh(0)519 217 y Fl(6\021)d Fr(b)670 232 y Fm(i)694 241 y Fh(0)800 217 y Ft(mo)s(d)g Fr(p)1069 180 y Fm(i)1093 189 y Fh(0)1132 217 y Ft(,)i(w)m(eil)i(b)s (eide)f(T)-8 b(erme)37 b(zwisc)m(hen)h(0)d(und)h Fr(p)2697 180 y Fm(i)2721 189 y Fh(0)2795 217 y Ft(liegen.)h(F)3169 221 y(\177)3166 217 y(ur)e(alle)h Fr(i)d Fl(\025)g Fr(i)3684 232 y Fn(0)3759 217 y Ft(gilt)0 333 y(dann)g(w)m(egen)h(\(2\))1351 466 y Fr(a)1402 481 y Fm(i)1458 466 y Fl(\021)29 b Fr(a)1615 481 y Fm(i)1639 490 y Fh(0)1705 466 y Fl(6\021)f Fr(b)1851 481 y Fm(i)1875 490 y Fh(0)1942 466 y Fl(\021)g Fr(b)2088 481 y Fm(i)2217 466 y Ft(mo)s(d)33 b Fr(p)2486 425 y Fm(i)2510 434 y Fh(0)0 617 y Ft(also)g Fr(a)247 632 y Fm(i)303 617 y Fl(6\021)28 b Fr(b)449 632 y Fm(i)544 617 y Ft(mo)s(d)33 b Fr(p)813 581 y Fm(i)837 590 y Fh(0)876 617 y Ft(.)f(Damit)h(ist)1618 748 y Fl(j)p Fr(a)1697 763 y Fm(i)1747 748 y Fl(\000)23 b Fr(b)1888 763 y Fm(i)1917 748 y Fl(j)1945 763 y Fm(p)2012 748 y Fr(>)k(p)2164 707 y Fi(\000)p Fm(i)2243 716 y Fh(0)0 899 y Ft(f)33 903 y(\177)30 899 y(ur)32 b(alle)h Fr(i)28 b Fl(\025)g Fr(i)532 914 y Fn(0)572 899 y Ft(,)33 b(also)f(\()p Fr(a)916 914 y Fm(i)945 899 y Ft(\))27 b Fl(6\030)h Ft(\()p Fr(b)1194 914 y Fm(i)1223 899 y Ft(\).)0 1027 y(Sei)i(n)m(un)f(eine)h(b)s (eliebige)h(Cauc)m(h)m(yfolge)f(\()p Fr(b)1562 1042 y Fm(i)1591 1027 y Ft(\))f(gegeb)s(en)h(\()p Fr(a)e Ft(=)p 2246 940 146 4 v 27 w(\()p Fr(b)2325 1042 y Fm(i)2354 1027 y Ft(\)\).)h(Wir)g(suc)m(hen)i(eine)f(dazu)3420 1031 y(\177)3420 1027 y(aquiv)-5 b(alen)m(te)0 1143 y(F)d(olge)46 b(\()p Fr(a)359 1158 y Fm(i)387 1143 y Ft(\),)h(die)g(\(1\))f(und)h (\(2\))f(erf)1335 1147 y(\177)1332 1143 y(ullt.)h(F)1619 1147 y(\177)1616 1143 y(ur)f(alle)h Fr(j)57 b Ft(=)52 b(1)p Fr(;)17 b Ft(2)p Fr(;)g Ft(3)p Fr(;)g(:)g(:)g(:)44 b Ft(sei)j Fr(n)2823 1158 y Fm(j)2907 1143 y Ft(die)g(nat)3222 1147 y(\177)3219 1143 y(urlic)m(he)h(Zahl,)e(so)0 1260 y(da\031)c Fl(j)p Fr(b)263 1275 y Fm(i)321 1260 y Fl(\000)29 b Fr(b)468 1275 y Fm(k)512 1260 y Fl(j)540 1275 y Fm(p)624 1260 y Fl(\024)46 b Fr(p)796 1223 y Fm(j)875 1260 y Ft(f)908 1264 y(\177)905 1260 y(ur)c(alle)h Fr(i;)17 b(k)49 b Fl(\025)c Fr(n)1585 1275 y Fm(j)1622 1260 y Ft(.)e(Wir)g(w)1970 1264 y(\177)1970 1260 y(ahlen)h(die)f Fr(n)2466 1275 y Fm(j)2545 1260 y Ft(streng)h(monoton)e(w)m(ac)m(hsend,)k(also)0 1376 y(insb)s(esondere)35 b Fr(n)631 1391 y Fm(j)695 1376 y Fl(\025)28 b Fr(j)6 b Ft(.)0 1492 y(Wir)33 b(hab)s(en)g Fl(j)p Fr(b)557 1507 y Fm(i)585 1492 y Fl(j)613 1507 y Fm(p)680 1492 y Fl(\024)28 b Ft(1)33 b(f)900 1496 y(\177)897 1492 y(ur)f(alle)h Fr(i)28 b Fl(\025)g Fr(n)1424 1507 y Fn(1)1464 1492 y Ft(,)k(denn)i(f)1795 1496 y(\177)1792 1492 y(ur)e(alle)h Fr(k)e Fl(\025)d Fr(n)2340 1507 y Fn(1)2412 1492 y Ft(gilt)984 1668 y Fl(j)p Fr(b)1053 1683 y Fm(i)1081 1668 y Fl(j)1109 1683 y Fm(p)1176 1668 y Fl(\024)h Ft(max\()p Fl(j)p Fr(b)1570 1683 y Fm(k)1613 1668 y Fl(j)1641 1683 y Fm(p)1680 1668 y Fr(;)17 b Fl(j)p Fr(b)1793 1683 y Fm(i)1843 1668 y Fl(\000)23 b Fr(b)1984 1683 y Fm(k)2027 1668 y Fl(j)2055 1683 y Fm(p)2095 1668 y Ft(\))k Fl(\024)h Ft(max)q(\()p Fl(j)p Fr(b)2554 1683 y Fm(k)2597 1668 y Fl(j)2625 1683 y Fm(p)2664 1668 y Fr(;)17 b(p)2757 1626 y Fi(\000)p Fn(1)2851 1668 y Ft(\))p Fr(;)0 1839 y Ft(und)33 b Fl(j)p Fr(b)264 1854 y Fm(k)307 1839 y Fl(j)335 1854 y Fm(p)402 1839 y Fl(\000)-60 b(!)28 b(j)p Fr(a)p Fl(j)654 1854 y Fm(p)721 1839 y Fl(\024)g Ft(1)k(mit)h Fr(k)e Fl(\000)-60 b(!)27 b(1)p Ft(.)0 1955 y(Wir)i(k)245 1959 y(\177)245 1955 y(onnen)h(daher)g(Lemma)f(19.6)g(an) m(w)m(enden)i(und)e(k)m(onstruieren)i(ganze)f(Zahlen)f Fr(a)3171 1970 y Fm(i)3228 1955 y Ft(mit)h(0)d Fl(\024)h Fr(a)3636 1970 y Fm(i)3692 1955 y Fr(<)g(p)3845 1919 y Fm(i)3873 1955 y Ft(,)0 2072 y(so)33 b(da\031)1599 2203 y Fl(j)p Fr(a)1678 2218 y Fm(i)1728 2203 y Fl(\000)22 b Fr(b)1868 2218 y Fm(n)1911 2228 y Fg(i)1942 2203 y Fl(j)1970 2218 y Fm(p)2037 2203 y Fl(\024)28 b Fr(p)2191 2161 y Fi(\000)p Fm(i)2274 2203 y Fr(:)0 2354 y Ft(Wir)33 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Ft(=)28 b(\()p Fr(a)2022 5338 y Fm(n)2069 5323 y Fr(a)2120 5338 y Fm(n)p Fi(\000)p Fn(1)2274 5323 y Fr(:)17 b(:)g(:)f(a)2456 5338 y Fn(1)2496 5323 y Fr(a)2547 5338 y Fn(0)2586 5323 y Ft(\))2624 5338 y Fm(p)2664 5323 y Fr(:)p eop end %%Page: 39 39 TeXDict begin 39 38 bop 1580 -99 a Fq(Die)26 b Fb(p)p Fq(-adisc)n(hen)f(Zahlen)1503 b(39)0 100 y Ft(W)-8 b(egen)38 b(\(2\))e Fr(b)521 115 y Fm(n)p Fi(\000)p Fn(1)694 100 y Fl(\021)f Fr(b)847 115 y Fm(n)961 100 y Ft(mo)s(d)e Fr(p)1230 64 y Fm(n)1314 100 y Ft(un)m(tersc)m(heiden)40 b(sic)m(h)e Fr(b)2169 115 y Fm(n)p Fi(\000)p Fn(1)2343 100 y Ft(und)g Fr(b)2584 115 y Fm(n)2668 100 y Ft(um)f(ein)h(Vielfac)m (hes)g(v)m(on)g Fr(p)3702 64 y Fm(n)3749 100 y Ft(,)f(so)0 217 y(da\031)31 b(b)s(ei)h(der)f(Darstellung)h(v)m(on)g Fr(b)1256 232 y Fm(n)1334 217 y Ft(dieselb)s(en)i(Zi\013ern)d Fr(a)2115 232 y Fm(n)p Fi(\000)p Fn(1)2269 217 y Fr(:)17 b(:)g(:)f(a)2451 232 y Fn(1)2491 217 y Fr(a)2542 232 y Fn(0)2613 217 y Ft(v)m(erw)m(endet)34 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y Fr(a)2079 1242 y Fn(1)2119 1227 y Fr(a)2170 1242 y Fn(0)2209 1227 y Ft(\))2247 1242 y Fm(p)2287 1227 y Fr(:)3823 1397 y Fd(\003)0 1579 y Fp(Bemerkung)50 b(19.9)e Ft(\(Die)41 b(Darstellung)h(v)m(on)g Fr(p)p Ft(-adisc)m(hen)g(Zahlen\))p Fp(.)47 b Ft(W)-8 b(enn)42 b Fr(a)f Ft(eine)i Fr(p)p Ft(-adisc)m(he)f(Zahl)0 1695 y(in)d Fk(Q)197 1710 y Fm(p)263 1695 y Fl(n)26 b Fk(Z)405 1710 y Fm(p)483 1695 y Ft(ist,)39 b(dann)f(ist)h Fl(j)p Fr(a)p Fl(j)1150 1710 y Fm(p)1227 1695 y Ft(=)e Fr(p)1389 1659 y Fm(n)1474 1695 y Ft(f)1507 1699 y(\177)1504 1695 y(ur)h(ein)h Fr(n)f(>)f Ft(0.)h(Wir)g(m)m(ultiplizieren)j Fr(a)e Ft(mit)f Fr(p)3267 1659 y Fm(n)3352 1695 y Ft(und)h(erhalten)0 1819 y Fr(b)d Ft(=)f Fr(p)237 1782 y Fm(n)284 1819 y Fr(a)i Ft(mit)g Fl(j)p Fr(b)p Fl(j)652 1834 y Fm(p)727 1819 y Fl(\024)e Ft(1.)i(Dann)g(l)1248 1823 y(\177)1248 1819 y(a\031t)f(sic)m(h)j Fr(b)e Ft(sc)m(hreib)s(en)i(als)e Fr(b)f Ft(=)f(\()p Fr(:)17 b(:)g(:)f(a)2688 1834 y Fn(2)2728 1819 y Fr(a)2779 1834 y Fn(1)2818 1819 y Fr(a)2869 1834 y Fn(0)2909 1819 y Ft(\))2947 1834 y Fm(p)3024 1819 y Ft(o)s(der)36 b(als)3411 1797 y(\177)3399 1819 y(Aquiv)-5 b(alenz-)0 1942 y(klasse)28 b(der)g(T)-8 b(eilsummenfolge)29 b(\()1212 1867 y Fj(P)1317 1894 y Fm(m)1317 1971 y(k)r Fn(=0)1467 1942 y Fr(a)1518 1957 y Fm(k)1561 1942 y Fr(p)1610 1906 y Fm(k)1652 1942 y Ft(\).)e(Dann)f(l)2029 1946 y(\177)2029 1942 y(a\031t)h(sic)m(h)h Fr(a)f Ft(sc)m(hreib)s(en)i(als)3033 1921 y(\177)3021 1942 y(Aquiv)-5 b(alenzklasse)31 b(der)0 2062 y(T)-8 b(eilsummenfolge)37 b(\()783 1987 y Fj(P)888 2013 y Fm(m)888 2091 y(k)r Fn(=0)1021 2062 y Ft(\()p Fr(a)1110 2077 y Fm(k)1176 2062 y Fl(\001)24 b Fr(p)1277 2026 y Fi(\000)p Fm(n)1378 2062 y Ft(\))p Fr(p)1465 2026 y Fm(k)1508 2062 y Ft(\))31 b(=)g(\()1722 1987 y Fj(P)1827 2013 y Fm(m)1827 2091 y(k)r Fn(=)p Fi(\000)p Fm(n)2039 2062 y Fr(a)2090 2077 y Fm(k)r Fn(+)p Fm(n)2231 2062 y Fr(p)2280 2026 y Fm(k)2322 2062 y Ft(\))h(=)f(\()2537 1987 y Fj(P)2642 2013 y Fm(m)2642 2091 y(k)r Fn(=)p Fi(\000)p Fm(n)2854 2062 y Fr(a)2905 2026 y Fi(0)2905 2088 y Fm(k)2948 2062 y Fr(p)2997 2026 y Fm(k)3039 2062 y Ft(\))k(mit)g Fr(a)3344 2026 y Fi(0)3344 2088 y Fm(k)3418 2062 y Ft(:=)c Fr(a)3603 2077 y Fm(k)r Fn(+)p Fm(n)3778 2062 y Ft(f)3811 2066 y(\177)3808 2062 y(ur)0 2178 y(alle)i Fr(k)e Fl(\025)d(\000)p Fr(n)p Ft(.)33 b(Die)g Fr(p)p Ft(-adisc)m(he)g(Zahl)f Fr(a)h Ft(hat)g(damit)f(eine)i(eindeutige)g Fr(p)p Ft(-adisc)m(he)g (Darstellung)1252 2348 y Fr(a)28 b Ft(=)f(\()p Fr(:)17 b(:)g(:)f(a)1654 2363 y Fn(2)1694 2348 y Fr(a)1745 2363 y Fn(1)1785 2348 y Fr(a)1836 2363 y Fn(0)1875 2348 y Fr(;)50 b(a)2003 2363 y Fi(\000)p Fn(1)2097 2348 y Fr(a)2148 2363 y Fi(\000)p Fn(2)2259 2348 y Fr(:)17 b(:)g(:)f(a)2441 2363 y Fi(\000)p Fm(n)2543 2348 y Ft(\))2581 2363 y Fm(p)2621 2348 y Fr(:)0 2518 y Ft(Sie)23 b(k)-5 b(ann)23 b(mit)h(einem)g (unendlic)m(hen)h(Dezimalbruc)m(h)g(v)m(erglic)m(hen)g(w)m(erden.)f (Die)2920 2578 y(")2965 2518 y(kleinen)-7 b(\\)31 b(T)-8 b(erme)24 b(stehen)0 2635 y(in)33 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