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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. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 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 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
readonly def
/FontBBox{-36 -250 1070 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 2 /multiply put
dup 106 /bar put
readonly def
/FontBBox{-30 -955 1185 779}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. 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 88 /summationdisplay put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
currentdict end
currentfile eexec
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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 100 /d put
dup 110 /n put
dup 112 /p put
readonly def
/FontBBox{-24 -250 1110 750}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 70 /F put
dup 90 /Z put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
currentdict end
currentfile eexec
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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 18 /theta put
dup 30 /phi put
dup 58 /period put
dup 59 /comma put
dup 61 /slash put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 78 /N put
dup 80 /P put
dup 81 /Q put
dup 82 /R put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 109 /m put
dup 110 /n put
dup 112 /p put
dup 116 /t put
dup 120 /x put
dup 121 /y put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
currentdict end
currentfile eexec
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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 12 /fi 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 61 /equal put
dup 63 /question put
dup 66 /B 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 76 /L put
dup 77 /M put
dup 83 /S put
dup 84 /T put
dup 91 /bracketleft put
dup 93 /bracketright 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 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 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 123 /endash put
readonly def
/FontBBox{-34 -251 988 750}readonly def
currentdict end
currentfile eexec
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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 58 /colon put
dup 65 /A put
dup 80 /P put
dup 101 /e put
dup 110 /n put
dup 114 /r put
dup 115 /s put
dup 119 /w put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
currentdict end
currentfile eexec
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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 12 /fi put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four 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 65 /A put
dup 66 /B put
dup 70 /F put
dup 72 /H put
dup 73 /I put
dup 80 /P put
dup 82 /R put
dup 84 /T put
dup 87 /W 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 121 /y put
dup 122 /z put
readonly def
/FontBBox{-36 -251 1103 750}readonly def
currentdict end
currentfile eexec
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%%BeginFont: CMR17
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%%CreationDate: 1991 Aug 20 16:38:24
% 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 (CMR17) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR17 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
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dup 41 /parenright put
dup 51 /three put
dup 52 /four put
dup 55 /seven put
dup 67 /C put
dup 80 /P put
dup 83 /S put
dup 97 /a put
dup 101 /e put
dup 103 /g put
dup 104 /h put
dup 108 /l put
dup 109 /m 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 121 /y put
readonly def
/FontBBox{-33 -250 945 749}readonly def
currentdict end
currentfile eexec
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TeXDict begin 39139632 55387786 1000 600 600 (sample3-374-07.dvi)
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rf /Fe 167[144 88[{}1 99.6264 /CMEX10 rf /Ff 143[36 1[43
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%%EndProlog
%%BeginSetup
%%Feature: *Resolution 600dpi
TeXDict begin
%%PaperSize: A4
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userdict/PStoPSxform PStoPSmatrix matrix currentmatrix
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%%Page: (0,1) 1
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%%BeginDocument: tcdarms.ps
%!
%%Title: tcdarms.fig
%%Creator: f2ps
%%CreationDate: Mon Jul  2 11:43:30 1990
%%For: dwilkins@lanczos (David Wilkins,,,1948,1229)
%%Pages: 1
%%BoundingBox: 89 162 523 630
%%EndComments
/$F2psDict 32 dict def $F2psDict begin $F2psDict /mtrx matrix put
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%%EndProlog
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 stroke
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 stroke
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 stroke
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 stroke
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 stroke
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 stroke
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 stroke
newpath 298.500 407.750 65.252 69.829 110.171 arc
 stroke
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 stroke
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 stroke
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 stroke
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 stroke
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 stroke
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 stroke
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 stroke
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 stroke
% Ellipse
newpath 299 263 13 6 0 360 DrawEllipse
 stroke
% Ellipse
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 stroke
% Polyline
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% Polyline
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% Polyline
newpath 349 250 moveto 359 260 lineto stroke
% Polyline
newpath 359 280 moveto 379 260 lineto stroke
% Polyline
newpath 359 300 moveto 399 260 lineto stroke
% Polyline
newpath 359 320 moveto 419 260 lineto stroke
% Polyline
newpath 367 332 moveto 439 260 lineto stroke
% Polyline
newpath 359 360 moveto 439 280 lineto stroke
% Polyline
newpath 379 360 moveto 439 300 lineto stroke
% Polyline
newpath 399 360 moveto 439 320 lineto stroke
% Polyline
newpath 419 359 moveto 439 340 lineto stroke
% Polyline
newpath 419 260 moveto 439 280 lineto stroke
% Polyline
newpath 399 260 moveto 439 300 lineto stroke
% Polyline
newpath 379 260 moveto 439 320 lineto stroke
% Polyline
newpath 359 260 moveto 439 340 lineto stroke
% Polyline
newpath 367 288 moveto 439 360 lineto stroke
% Polyline
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% Polyline
newpath 359 320 moveto 399 360 lineto stroke
% Polyline
newpath 359 340 moveto 379 360 lineto stroke
% Polyline
newpath 529 143 moveto 542 169 lineto stroke
% Polyline
newpath 489 139 moveto 518 198 lineto stroke
% Polyline
newpath 495 142 moveto 522 193 lineto stroke
% Polyline
newpath 540 141 moveto 550 161 lineto stroke
% Polyline
newpath 534 142 moveto 546 165 lineto stroke
% Polyline
newpath 522 143 moveto 538 173 lineto stroke
% Polyline
newpath 515 143 moveto 534 178 lineto stroke
% Polyline
newpath 509 143 moveto 530 184 lineto stroke
% Polyline
newpath 502 143 moveto 526 189 lineto stroke
% Polyline
newpath 482 138 moveto 514 204 lineto stroke
% Polyline
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% Polyline
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% Polyline
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3.000 setlinewidth
% Polyline
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1.000 setlinewidth
% Polyline
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% Polyline
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3.000 setlinewidth
% Polyline
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% Polyline
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% Polyline
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1.000 setlinewidth
% Polyline
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% Polyline
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% Polyline
newpath 487 418 moveto 487 432 lineto stroke
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
newpath 324 457 moveto 324 469 lineto stroke
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
newpath 312 471 moveto 312 486 lineto stroke
% Polyline
newpath 299 460 moveto 299 472 lineto stroke
% Polyline
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% Polyline
newpath 312 444 moveto 312 458 lineto stroke
% Polyline
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% Polyline
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% Polyline
newpath 312 418 moveto 312 432 lineto stroke
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
newpath 512 282 moveto 539 301 lineto 567 282 lineto stroke
% Polyline
newpath 576 288 moveto 548 307 lineto 576 325 lineto stroke
% Polyline
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% Polyline
newpath 221 303 moveto 255 303 lineto 255 282 lineto stroke
% Polyline
newpath 262 332 moveto 262 310 lineto 296 310 lineto stroke
% Polyline
newpath 221 310 moveto 255 310 lineto 255 332 lineto stroke
% Polyline
newpath 221 282 moveto 221 331 lineto 296 331 lineto 296 282 lineto closepath  stroke
% Polyline
newpath 430 469 moveto 474 469 lineto stroke
% Polyline
newpath 368 469 moveto 321 469 lineto stroke
% Polyline
newpath 430 457 moveto 477 457 lineto stroke
% Polyline
newpath 368 457 moveto 321 457 lineto stroke
% Polyline
newpath 370 444 moveto 280 444 lineto stroke
% Polyline
newpath 381 432 moveto 280 432 lineto stroke
% Polyline
newpath 415 419 moveto 415 430 lineto stroke
% Polyline
newpath 387 419 moveto 387 428 lineto stroke
% Polyline
newpath 402 432 moveto 408 432 lineto stroke
% Polyline
newpath 396 432 moveto 390 432 lineto stroke
% Polyline
newpath 383 438 moveto 415 438 lineto stroke
% Polyline
newpath 378 444 moveto 420 444 lineto stroke
% Polyline
newpath 375 450 moveto 423 450 lineto stroke
% Polyline
newpath 417 455 moveto 418 457 lineto 420 455 lineto stroke
% Polyline
newpath 410 455 moveto 412 457 lineto 413 455 lineto stroke
% Polyline
newpath 404 455 moveto 405 457 lineto 407 455 lineto stroke
% Polyline
newpath 398 455 moveto 399 457 lineto 401 455 lineto stroke
% Polyline
newpath 392 455 moveto 393 457 lineto 395 455 lineto stroke
% Polyline
newpath 385 455 moveto 387 457 lineto 388 455 lineto stroke
% Polyline
newpath 379 455 moveto 380 457 lineto 382 455 lineto stroke
% Polyline
newpath 418 440 moveto 418 457 lineto stroke
% Polyline
newpath 412 435 moveto 412 457 lineto stroke
% Polyline
newpath 405 433 moveto 405 457 lineto stroke
% Polyline
newpath 380 441 moveto 380 457 lineto stroke
% Polyline
newpath 387 435 moveto 387 457 lineto stroke
% Polyline
newpath 393 433 moveto 393 457 lineto stroke
% Polyline
newpath 399 432 moveto 399 457 lineto stroke
% Polyline
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% Polyline
newpath 422 447 moveto 428 445 lineto 423 437 lineto 418 441 lineto stroke
% Polyline
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% Polyline
newpath 374 460 moveto 368 460 lineto 368 453 lineto 374 453 lineto stroke
% Polyline
newpath 377 447 moveto 370 444 lineto 375 437 lineto 380 441 lineto stroke
% Polyline
newpath 384 437 moveto 381 431 lineto 388 427 lineto 390 433 lineto stroke
% Polyline
newpath 374 457 moveto 374 482 lineto stroke
% Polyline
newpath 424 457 moveto 424 482 lineto stroke
% Polyline
newpath 396 432 moveto 395 426 lineto 403 426 lineto 402 432 lineto stroke
% Polyline
newpath 374 472 moveto 368 472 lineto 368 466 lineto 374 466 lineto stroke
% Polyline
newpath 424 472 moveto 430 472 lineto 430 466 lineto 424 466 lineto stroke
% Polyline
newpath 312 388 moveto 324 388 lineto 324 397 lineto 343 397 lineto 343 388 lineto 368 388 lineto 368 397 lineto 387 397 lineto 387 388 lineto 412 388 lineto 412 397 lineto 430 397 lineto 430 388 lineto 455 388 lineto 455 397 lineto 474 397 lineto 474 388 lineto 487 388 lineto stroke
% Polyline
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% Polyline
newpath 349 407 moveto 349 419 lineto stroke
% Polyline
newpath 374 407 moveto 374 419 lineto stroke
% Polyline
newpath 337 419 moveto 337 432 lineto stroke
% Polyline
newpath 362 419 moveto 362 432 lineto stroke
% Polyline
newpath 349 457 moveto 349 469 lineto stroke
% Polyline
newpath 349 432 moveto 349 444 lineto stroke
% Polyline
newpath 337 444 moveto 337 457 lineto stroke
% Polyline
newpath 362 444 moveto 362 457 lineto stroke
% Polyline
newpath 337 469 moveto 337 482 lineto stroke
% Polyline
newpath 362 469 moveto 362 482 lineto stroke
% Polyline
newpath 315 419 moveto 399 419 lineto stroke
% Polyline
newpath 315 407 moveto 483 407 lineto stroke
% Polyline
newpath 312 388 moveto 324 482 lineto 474 482 lineto 486 388 lineto stroke
% Polyline
newpath 480 419 moveto 399 419 lineto stroke
% Polyline
newpath 437 469 moveto 437 482 lineto stroke
% Polyline
newpath 462 469 moveto 462 481 lineto stroke
% Polyline
newpath 437 444 moveto 437 457 lineto stroke
% Polyline
newpath 462 444 moveto 462 457 lineto stroke
% Polyline
newpath 449 432 moveto 449 444 lineto stroke
% Polyline
newpath 449 457 moveto 449 469 lineto stroke
% Polyline
newpath 437 419 moveto 437 432 lineto stroke
% Polyline
newpath 462 419 moveto 462 432 lineto stroke
% Polyline
newpath 424 407 moveto 424 419 lineto stroke
% Polyline
newpath 449 407 moveto 449 419 lineto stroke
% Polyline
newpath 496 269 moveto 496 363 lineto 502 363 lineto 502 269 lineto closepath  stroke
% Polyline
newpath 374 432 moveto 374 438 lineto stroke
% Polyline
newpath 424 432 moveto 424 438 lineto stroke
% Polyline
newpath 315 403 moveto 485 403 lineto stroke
% Polyline
newpath 399 419 moveto 399 407 lineto stroke
% Polyline
newpath 427 444 moveto 518 444 lineto stroke
% Polyline
newpath 418 432 moveto 518 432 lineto stroke
% Interpolated spline
newpath 482 221 moveto
	479.025 211.731 477.683 207.699 477 205 curveto
	475.514 202.455 472.296 196.768 471 194 curveto
	470.153 191.845 467.955 186.007 467 184 curveto
	466.423 181.080 465.244 175.170 464 173 curveto
	463.907 170.870 462.901 168.182 460 162 curveto
 stroke
% Interpolated spline
newpath 471 146 moveto
	471.085 152.096 471.085 154.783 471 157 curveto
	471.354 159.100 472.087 164.938 473 167 curveto
	473.006 169.867 474.614 175.676 475 178 curveto
	475.836 180.574 476.661 186.487 478 189 curveto
	479.516 191.953 484.762 197.461 487 200 curveto
	489.552 201.545 495.417 205.050 498 206 curveto
	499.810 207.223 502.494 208.231 509 210 curveto
 stroke
% Interpolated spline
newpath 460 145 moveto
	460.467 151.535 460.467 154.559 460 157 curveto
	460.467 160.299 460.162 169.059 460 173 curveto
	460.776 176.492 462.176 185.237 463 189 curveto
	463.834 191.430 466.128 197.220 467 200 curveto
	468.545 202.773 471.823 210.200 474 213 curveto
	475.447 215.235 479.585 219.671 482 221 curveto
	485.435 223.292 493.798 226.337 498 227 curveto
	500.104 226.580 502.787 225.908 509 224 curveto
 stroke
% Interpolated spline
newpath 474 135 moveto
	490.729 140.462 498.108 142.478 503 143 curveto
	507.113 143.685 515.828 143.340 519 143 curveto
	523.185 143.034 531.940 142.451 536 142 curveto
	540.612 141.013 552.176 137.693 557 136 curveto
	558.289 136.163 561.160 135.282 562 135 curveto
	563.669 134.973 566.324 133.647 568 135 curveto
	568.907 136.234 568.145 138.172 568 139 curveto
	567.248 140.732 565.906 142.412 562 146 curveto
 stroke
% Interpolated spline
newpath 466 136 moveto
	467.315 135.586 467.986 135.251 469 135 curveto
	469.783 134.826 472.546 134.570 474 135 curveto
	474.829 135.517 476.175 136.866 477 138 curveto
	477.120 139.152 477.120 141.848 477 143 curveto
	476.175 144.135 474.829 145.483 474 146 curveto
	471.209 146.984 465.824 146.984 463 146 curveto
	461.259 145.091 458.826 141.949 458 141 curveto
	456.959 139.355 455.271 136.690 455 135 curveto
	454.942 133.682 455.675 130.947 456 130 curveto
	457.633 127.891 461.052 125.112 463 124 curveto
	466.099 123.339 472.425 123.920 475 124 curveto
	478.176 124.855 484.511 127.593 487 128 curveto
	490.936 129.467 499.580 131.966 503 132 curveto
	507.106 132.894 515.828 132.590 519 132 curveto
	523.185 132.284 531.940 131.701 536 131 curveto
	540.612 130.263 552.176 126.943 557 126 curveto
	558.289 125.413 561.160 124.532 562 124 curveto
	564.286 124.147 568.641 124.222 570 124 curveto
	572.351 124.530 576.690 124.769 579 126 curveto
	580.248 126.602 583.117 129.200 584 131 curveto
	584.454 132.433 584.344 135.160 584 136 curveto
	582.534 140.492 579.180 144.859 570 154 curveto
 stroke
% Interpolated spline
newpath 239 136 moveto
	241.420 136.709 242.405 137.062 243 138 curveto
	243.585 138.160 244.019 139.535 244 140 curveto
	244.395 141.314 244.201 143.895 244 145 curveto
	244.124 145.766 244.805 147.521 244 148 curveto
	243.064 149.129 242.080 149.129 241 148 curveto
 stroke
% Interpolated spline
newpath 233 168 moveto
	223.338 166.674 219.190 165.967 217 164 curveto
	210.952 160.908 200.412 147.422 202 145 curveto
	202.624 144.835 205.803 148.135 207 147 curveto
	208.034 146.118 200.600 138.671 202 137 curveto
	202.749 136.599 204.672 139.025 206 138 curveto
	207.666 136.651 201.948 130.273 203 127 curveto
	203.385 126.055 204.198 124.871 206 125 curveto
	207.251 125.034 207.012 128.725 209 129 curveto
	210.244 128.498 210.105 125.535 211 125 curveto
	212.118 124.475 213.831 124.010 215 125 curveto
	217.366 128.118 212.662 133.674 214 137 curveto
	213.296 140.224 215.943 147.810 218 151 curveto
	219.131 152.633 222.539 155.497 225 156 curveto
	225.702 157.560 229.350 159.387 232 160 curveto
	232.199 159.834 233.535 159.692 237 160 curveto
 stroke
% Interpolated spline
newpath 245 142 moveto
	241.388 140.931 240.052 140.860 239 142 curveto
	237.921 141.249 234.550 142.040 235 143 curveto
	234.641 144.069 238.439 144.470 239 145 curveto
	240.428 144.858 241.553 144.929 244 145 curveto
 stroke
% Interpolated spline
newpath 251 153 moveto
	244.613 152.473 241.871 152.473 241 153 curveto
	238.045 153.318 232.857 154.033 233 156 curveto
	232.354 158.938 240.210 158.820 241 162 curveto
	240.231 166.119 229.432 166.975 230 170 curveto
	230.405 170.789 235.456 168.941 236 171 curveto
	236.818 172.630 229.251 175.331 230 178 curveto
	231.546 180.411 241.888 178.151 246 178 curveto
	246.938 176.837 249.689 175.737 251 174 curveto
	251.269 173.431 249.962 171.276 252 171 curveto
	252.869 169.232 255.190 172.737 256 172 curveto
	258.958 169.388 253.431 163.282 256 162 curveto
	256.809 160.850 258.870 164.445 261 164 curveto
	262.756 162.054 258.117 156.523 261 154 curveto
	262.218 152.640 264.644 155.168 266 154 curveto
	268.927 151.808 266.168 146.119 269 144 curveto
	269.898 142.689 272.708 144.536 274 143 curveto
	276.394 140.081 275.283 135.238 274 133 curveto
	272.992 131.394 270.839 130.578 270 130 curveto
	263.467 128.697 249.840 128.842 244 131 curveto
	242.290 131.288 239.817 132.517 239 134 curveto
	238.422 135.323 238.492 137.018 239 140 curveto
 stroke
% Interpolated spline
newpath 248 177 moveto
	248.638 180.810 248.638 182.857 248 185 curveto
	248.689 186.833 250.270 192.642 250 196 curveto
	247.991 197.629 244.641 199.896 243 200 curveto
	239.617 201.880 232.542 201.285 229 201 curveto
	227.737 201.612 222.498 200.171 221 203 curveto
	220.673 204.256 226.981 206.554 227 209 curveto
	227.494 211.418 223.454 213.897 225 215 curveto
	227.673 219.862 242.611 209.020 245 213 curveto
	245.397 213.746 242.785 215.643 243 216 curveto
	244.530 217.443 257.309 208.991 261 205 curveto
	262.336 201.552 262.562 194.969 263 192 curveto
	263.380 189.731 260.982 184.083 264 181 curveto
	266.517 179.415 270.416 181.943 272 182 curveto
	278.314 182.174 292.688 182.103 299 182 curveto
	305.308 181.291 320.660 181.811 327 179 curveto
	328.383 177.820 328.836 175.092 331 176 curveto
	331.974 175.376 332.237 178.785 333 180 curveto
	337.174 184.705 351.608 189.413 354 198 curveto
	353.865 199.771 352.833 201.547 352 203 curveto
	348.852 204.833 339.120 202.439 338 205 curveto
	338.453 206.358 343.171 206.028 344 208 curveto
	344.325 210.748 339.468 212.397 341 214 curveto
	342.559 217.773 352.862 215.316 358 214 curveto
	358.941 212.813 361.740 209.612 363 208 curveto
	364.858 205.022 370.025 199.186 369 195 curveto
	367.131 189.813 360.009 189.089 358 187 curveto
	354.859 184.450 348.913 177.729 347 174 curveto
	345.682 171.260 344.448 164.351 344 162 curveto
	343.681 159.251 341.299 154.749 344 152 curveto
	345.011 150.068 348.658 151.559 351 151 curveto
	351.269 150.473 354.359 149.064 354 148 curveto
	353.547 145.715 345.064 147.920 343 146 curveto
	341.117 145.744 340.141 143.104 340 143 curveto
	336.963 141.371 332.429 141.525 331 142 curveto
	328.133 141.063 322.626 143.568 320 140 curveto
	318.308 138.286 318.797 134.922 322 134 curveto
	324.791 129.891 332.920 136.492 334 134 curveto
	334.435 133.403 331.915 132.450 333 131 curveto
	332.550 130.731 335.200 131.157 336 131 curveto
	340.937 131.101 355.484 136.521 356 134 curveto
	356.940 133.004 352.719 132.291 353 130 curveto
	352.719 128.874 356.950 128.295 356 128 curveto
	355.996 125.632 347.971 127.752 345 128 curveto
	342.264 127.765 334.790 127.244 332 128 curveto
	328.090 127.733 320.033 126.725 317 130 curveto
	313.857 132.380 316.497 138.417 314 140 curveto
	312.088 142.002 309.209 140.117 308 142 curveto
	306.576 141.738 304.849 143.758 306 145 curveto
	307.134 148.299 314.805 146.094 318 146 curveto
	320.573 146.739 327.494 144.724 331 148 curveto
	332.405 149.342 333.380 152.242 332 154 curveto
	324.235 165.087 306.016 159.209 298 160 curveto
	294.725 159.886 286.646 159.222 283 160 curveto
	279.454 158.995 273.900 158.854 261 159 curveto
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showpage
$F2psEnd

%%EndDocument
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b(Timoth)m(y)e(Murph)m(y)605 2674 y(GMB)i(??)400 b(F)-10
b(rida)m(y)g(,)38 b(??)52 b(2007)346 b(??:00{??:00)501
3049 y Fk(A)n(ttempt)36 b(4)e(questions)h(fr)-5 b(om)34
b(Part)h(A,)g(and)g(2)f(questions)h(fr)-5 b(om)34 b(Part)h(B.)1725
3277 y Fj(P)p Fi(art)d(B)377 3505 y(7.)48 b(Let)38 b(the)g(n)m(um)m(b)s
(er)g(of)f(irreducible)j(p)s(olynomials)e(of)f(degree)h
Fh(d)f Fi(o)m(v)m(er)i Fg(F)3151 3520 y Ff(p)3228 3505
y Fi(b)s(e)e(de-)501 3626 y(noted)c(b)m(y)h Fh(N)10 b
Fi(\()p Fh(d)p Fi(\).)43 b(Sho)m(w)34 b(that)1658 3863
y Fh(p)1707 3822 y Ff(n)1782 3863 y Fi(=)1885 3768 y
Fe(X)1908 3984 y Ff(d)p Fd(j)p Ff(n)2046 3863 y Fh(dN)10
b Fi(\()p Fh(d)p Fi(\))p Fh(:)501 4229 y Fi(Hence)36
b(or)e(otherwise)i(sho)m(w)g(that)f(there)g(is)g(at)f(least)h(one)g
(irreducible)h(p)s(olyno-)501 4350 y(mial)d(of)f(eac)m(h)i(degree)f
Fh(d)p Fi(.)501 4512 y(Ho)m(w)f(man)m(y)g(irreducible)h(p)s(olynomials)
f(are)f(there)h(of)f(degree)h(6)f(o)m(v)m(er)h Fg(F)3177
4527 y Fc(2)3217 4512 y Fi(?)43 b(Ho)m(w)501 4632 y(man)m(y)34
b(of)e(these)i(p)s(olynomials)f(are)g(primitiv)m(e?)501
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4794 y Fi(\).)501 4956 y Fj(Answ)m(er:)538 5184 y Fk(\(a\))1852
5637 y Fi(1)p eop end
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b(ho)m(w)f(p)s(oin)m(ts)g(on)g(an)f(elliptic)i(curv)m(e)g(are)f(added.)
501 735 y(Sho)m(w)h(that)1668 855 y Fh(y)1720 814 y Fc(2)1786
855 y Fi(=)27 b Fh(x)1944 814 y Fc(3)2006 855 y Fi(+)22
b Fh(x)h Fi(+)f(1)501 1029 y(de\014nes)41 b(an)e(elliptic)i(curv)m(e)g
(o)m(v)m(er)f Fg(F)1840 1044 y Fc(11)1915 1029 y Fi(,)h(and)e(\014nd)h
(the)f(order)h(of)f(the)g(group)g(on)501 1150 y(the)33
b(curv)m(e.)501 1312 y(Find)g(p)s(oin)m(ts)g(on)g(the)g(curv)m(e)h(of)e
(eac)m(h)h(p)s(ossible)h(order.)501 1474 y Fj(Answ)m(er:)538
1677 y Fk(\(a\))48 b(The)37 b(quadr)-5 b(atic)37 b(r)-5
b(esidues)40 b Fi(mo)s(d)t(11)d Fk(ar)-5 b(e)37 b Fi(0)p
Fh(;)17 b Fi(1)p Fh(;)g Fi(4)p Fh(;)g Fb(\000)p Fi(2)p
Fh(;)g Fi(5)p Fh(;)g Fi(3)p Fk(.)50 b(We)38 b(dr)-5 b(aw)37
b(up)g(a)716 1797 y(table)d(showing)g Fh(x;)45 b(x)1498
1761 y Fc(3)1560 1797 y Fi(+)22 b Fh(x)h Fi(+)f(1)34
b Fk(and)g(the)h(p)-5 b(ossible)34 b(values)g(for)h Fh(y)t
Fk(:)1684 2009 y Fh(x)p 1814 2045 4 121 v 118 w(x)1912
1973 y Fc(3)1974 2009 y Fi(+)22 b Fh(x)h Fi(+)f(1)149
b Fh(y)p 1607 2048 998 4 v 1687 2133 a Fi(0)p 1814 2169
4 121 v 317 w(1)306 b Fb(\006)p Fi(1)1687 2253 y(1)p
1814 2289 V 317 w(3)g Fb(\006)p Fi(5)1687 2373 y(2)p
1814 2410 V 317 w(0)345 b(0)1687 2494 y(3)p 1814 2530
V 278 w Fb(\000)p Fi(2)268 b Fb(\006)p Fi(3)1687 2614
y(4)p 1814 2650 V 317 w(3)306 b Fb(\006)p Fi(5)1687 2735
y(5)p 1814 2771 V 278 w Fb(\000)p Fi(1)293 b Fb(\000)1648
2855 y(\000)p Fi(5)p 1814 2891 V 279 w(3)278 b Fh(pm)p
Fi(5)1648 2975 y Fb(\000)p Fi(4)p 1814 3011 V 240 w Fb(\000)p
Fi(1)293 b Fb(\000)1648 3096 y(\000)p Fi(3)p 1814 3132
V 279 w(4)306 b Fb(\006)p Fi(2)1648 3216 y Fb(\000)p
Fi(2)p 1814 3252 V 279 w(2)331 b Fb(\000)1648 3337 y(\000)p
Fi(1)p 1814 3373 V 240 w Fb(\000)p Fi(1)293 b Fb(\000)716
3570 y Fk(T)-7 b(aking)33 b(the)h(p)-5 b(oint)34 b(at)g(in\014nity)g
(into)h(ac)-5 b(c)g(ount,)34 b(the)g(curve)g(has)g(14)g(p)-5
b(oints.)716 3691 y(It)35 b(fol)5 b(lows)35 b(that)g(the)h(gr)-5
b(oup)35 b(on)g(the)h(curve)f(is)g Fg(Z)p Fh(=)p Fi(\(14\))p
Fk(,)h(with)f(elements)g(of)716 3811 y(or)-5 b(der)34
b(1,2,7,14.)716 3952 y(We)f(know)f(that)i(if)f Fh(d)27
b Fb(j)g Fh(n)33 b Fk(then)g(ther)-5 b(e)33 b(ar)-5 b(e)33
b(just)h Fh(\036)p Fi(\()p Fh(d)p Fi(\))e Fk(elements)g(of)h(or)-5
b(der)33 b Fh(d)716 4073 y Fk(in)j Fg(Z)p Fh(=)p Fi(\()p
Fh(n)p Fi(\))p Fk(.)51 b(Thus)36 b(ther)-5 b(e)37 b(is)f(1)h(element)f
(of)g(or)-5 b(der)37 b(1,)g(1)f(element)g(of)h(or)-5
b(der)716 4193 y(2,)34 b(and)h(6)f(elements)g(e)-5 b(ach)35
b(of)f(or)-5 b(ders)35 b Fi(7)f Fk(and)g Fi(14)p Fk(.)716
4334 y(The)39 b(zer)-5 b(o)40 b(element)f Fi([0)p Fh(;)17
b Fi(1)p Fh(;)g Fi(0])39 b Fk(has)h(or)-5 b(der)39 b(1.)60
b(If)40 b Fh(P)50 b Fi(=)37 b(\()p Fh(x;)17 b(y)t Fi(\))39
b Fk(then)h Fb(\000)p Fh(P)51 b Fi(=)716 4455 y(\()p
Fb(\000)p Fh(x;)17 b(y)t Fi(\))p Fk(.)59 b(Thus)39 b
Fh(P)53 b Fk(has)39 b(or)-5 b(der)40 b(2)f(if)h(and)f(only)g(if)h
Fh(y)f Fi(=)e(0)p Fk(.)59 b(We)39 b(se)-5 b(e)40 b(fr)-5
b(om)716 4575 y(the)35 b(table)f(ab)-5 b(ove)34 b(that)i
Fh(A)27 b Fi(=)h(\(2)p Fh(;)17 b Fi(0\))34 b Fk(is)h(such)f(a)h(p)-5
b(oint.)716 4716 y(If)40 b Fh(P)54 b Fk(has)39 b(or)-5
b(der)41 b(7)f(then)g Fb(\000)p Fh(P)54 b Fk(has)40 b(or)-5
b(der)40 b(14;)j(and)d(sinc)-5 b(e)40 b(ther)-5 b(e)40
b(ar)-5 b(e)40 b(the)716 4837 y(same)j(numb)-5 b(er)43
b(of)g(p)-5 b(oints)43 b(of)h(or)-5 b(ders)43 b(7)g(and)g(14,)i(the)f
(c)-5 b(onverse)43 b(is)g(also)716 4957 y(true:)i(if)35
b Fh(P)48 b Fk(has)34 b(or)-5 b(der)35 b(14)g(then)f
Fb(\000)p Fh(P)49 b Fk(has)34 b(or)-5 b(der)35 b(7.)716
5098 y(T)-7 b(ake)31 b(the)h(p)-5 b(oint)32 b Fh(B)h
Fi(=)28 b(\(0)p Fh(;)17 b Fi(1\))p Fk(.)43 b(R)-5 b(e)g(c)g(al)5
b(l)31 b(that)i Fh(P)d Fi(+)16 b Fh(Q)g Fi(+)g Fh(R)30
b Fi(=)d(0)32 b Fk(if)g(the)h(p)-5 b(oints)716 5218 y
Fh(P)s(;)17 b(Q;)g(R)39 b Fk(on)f(the)h(curve)g(ar)-5
b(e)38 b(c)-5 b(ol)5 b(line)-5 b(ar.)55 b(Thus)39 b(the)f(tangent)h(at)
g Fh(B)44 b Fk(me)-5 b(ets)716 5339 y(the)35 b(curve)g(again)f(in)g
(the)h(p)-5 b(oint)35 b Fb(\000)p Fi(2)p Fh(B)5 b Fk(.)p
eop end
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TeXDict begin 3 2 bop 716 573 a Fk(We)35 b(have)1757
810 y Fi(2)p Fh(y)1870 743 y(dy)p 1868 787 107 4 v 1868
879 a(dx)2011 810 y Fi(=)27 b(3)p Fh(x)2218 769 y Fc(2)2280
810 y Fi(+)22 b(1)p Fh(;)716 1069 y Fk(ie)1823 1274 y
Fh(dy)p 1821 1318 V 1821 1409 a(dx)1964 1341 y Fi(=)2078
1274 y(3)p Fh(x)2182 1237 y Fc(2)2244 1274 y Fi(+)g(1)p
2078 1318 313 4 v 2184 1409 a(2)p Fh(y)716 1618 y Fk(In)34
b(p)-5 b(articular,)35 b(at)g Fh(B)1653 1875 y(m)28 b
Fi(=)1882 1808 y Fh(dy)p 1880 1852 107 4 v 1880 1943
a(dx)2023 1875 y Fi(=)g(1)p Fh(=)p Fi(2)f(=)h Fb(\000)p
Fi(5)p Fh(:)716 2113 y Fk(Thus)34 b(the)h(tangent)g(at)g
Fh(B)40 b Fk(is)1701 2326 y Fh(y)26 b Fb(\000)c Fi(1)28
b(=)f Fb(\000)p Fi(5\()p Fh(x)c Fb(\000)g Fi(0\))p Fh(;)716
2564 y Fk(ie)1826 2802 y Fh(y)30 b Fi(=)e Fb(\000)p Fi(5)p
Fh(x)23 b Fi(+)f(1)p Fh(:)716 3034 y Fk(This)34 b(me)-5
b(ets)34 b(the)h(curve)g(wher)-5 b(e)1598 3247 y Fi(\()p
Fh(mx)22 b Fi(+)g Fh(c)p Fi(\))1976 3206 y Fc(2)2043
3247 y Fi(=)28 b Fh(x)2202 3206 y Fc(3)2264 3247 y Fi(+)22
b Fh(x)h Fi(+)f(1)p Fh(:)716 3460 y Fk(If)34 b(the)h(r)-5
b(o)g(ots)35 b(of)f(this)h(ar)-5 b(e)35 b Fh(x)1740 3475
y Fc(0)1780 3460 y Fh(;)17 b(x)1879 3475 y Fc(1)1918
3460 y Fh(;)g(x)2017 3475 y Fc(2)2092 3460 y Fk(then)1702
3673 y Fh(x)1757 3688 y Fc(0)1819 3673 y Fi(+)22 b Fh(x)1972
3688 y Fc(1)2034 3673 y Fi(+)g Fh(x)2187 3688 y Fc(2)2254
3673 y Fi(=)28 b Fh(m)2443 3632 y Fc(2)2482 3673 y Fh(:)716
3886 y Fk(We)h(know)e(that)i(two)g(of)f(the)h(r)-5 b(o)g(ots,)29
b(say)g Fh(x)2245 3901 y Fc(0)2285 3886 y Fh(;)17 b(x)2384
3901 y Fc(1)2423 3886 y Fk(,)30 b(ar)-5 b(e)28 b Fi(0)p
Fh(;)17 b Fi(0)p Fk(.)42 b(Henc)-5 b(e)29 b(the)f(thir)-5
b(d)716 4006 y(r)g(o)g(ot)35 b(is)1514 4127 y Fh(x)28
b Fi(=)g Fh(m)1786 4086 y Fc(2)1847 4127 y Fb(\000)23
b Fi(2)p Fh(x)2051 4142 y Fc(0)2118 4127 y Fi(=)28 b(25)22
b Fb(\000)g Fi(0)28 b(=)f(3)p Fh(:)716 4298 y Fk(F)-7
b(r)i(om)33 b(the)i(e)-5 b(quation)35 b(for)g(the)f(tangent,)1785
4511 y Fh(y)c Fi(=)e Fb(\000)p Fi(14)g(=)f Fb(\000)p
Fi(3)p Fh(:)716 4743 y Fk(Thus)43 b(the)h(tangent)g(at)f
Fh(B)49 b Fk(me)-5 b(ets)44 b(the)f(curve)h(again)f(in)g(the)h(p)-5
b(oint)43 b Fh(C)51 b Fi(=)716 4864 y(\(3)p Fh(;)17 b
Fb(\000)p Fi(3\))p Fk(:)1888 4984 y Fi(2)p Fh(B)33 b
Fi(=)27 b Fb(\000)p Fh(C)r(:)716 5175 y Fk(Applying)35
b(the)f(same)g(ar)-5 b(gument)35 b(with)g Fh(C)42 b Fk(in)34
b(plac)-5 b(e)34 b(of)h Fh(B)5 b Fk(,)35 b(we)f(now)g(have)1719
5388 y Fh(m)28 b Fi(=)f Fb(\000)p Fi(28)p Fh(=)p Fi(6)g(=)h
Fb(\000)p Fi(1)p Fh(:)p eop end
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TeXDict begin 4 3 bop 716 573 a Fk(Thus)34 b(the)h(tangent)g(at)g
Fh(C)42 b Fk(is)1727 777 y Fh(y)25 b Fi(+)d(3)27 b(=)h
Fb(\000)p Fi(\()p Fh(x)23 b Fb(\000)g Fi(3\))p Fh(;)716
1005 y Fk(ie)1934 1234 y Fh(y)31 b Fi(=)d Fb(\000)p Fh(x:)716
1456 y Fk(This)34 b(me)-5 b(ets)34 b(the)h(curve)g(again)f(wher)-5
b(e)1604 1660 y Fh(x)28 b Fi(=)f(\()p Fb(\000)p Fi(1\))1992
1619 y Fc(2)2054 1660 y Fb(\000)22 b Fi(2)g Fb(\001)g
Fi(3)28 b(=)f Fb(\000)p Fi(5)p Fh(:)716 1864 y Fk(F)-7
b(r)i(om)33 b(the)i(e)-5 b(quation)35 b(for)g(the)f(tangent,)1976
2068 y Fh(y)d Fi(=)d(5)p Fh(:)716 2290 y Fk(Thus)43 b(the)g(tangent)h
(at)f Fh(C)50 b Fk(me)-5 b(ets)43 b(the)h(curve)f(again)g(in)g(the)g(p)
-5 b(oint)43 b Fh(D)j Fi(=)716 2410 y(\()p Fb(\000)p
Fi(5)p Fh(;)17 b Fi(5\))p Fk(:)1753 2531 y Fi(4)p Fh(B)33
b Fi(=)27 b Fb(\000)p Fi(2)p Fh(C)35 b Fi(=)28 b Fh(D)s(:)716
2717 y Fk(R)-5 b(ep)g(e)g(ating)34 b(yet)h(again,)f(at)h
Fh(D)1772 2920 y(m)28 b Fi(=)f(76)p Fh(=)p Fi(10)g(=)g(1)p
Fh(:)716 3124 y Fk(Thus)34 b(the)h(tangent)g(at)g Fh(D)j
Fk(is)1803 3328 y Fh(y)26 b Fb(\000)c Fi(5)28 b(=)f Fh(x)c
Fi(+)f(5)p Fh(;)716 3557 y Fk(ie)1888 3785 y Fh(y)31
b Fi(=)c Fh(x)c Fb(\000)f Fi(1)p Fh(:)716 4008 y Fk(This)34
b(me)-5 b(ets)34 b(the)h(curve)g(again)f(wher)-5 b(e)1720
4211 y Fh(x)28 b Fi(=)f(1)1955 4170 y Fc(2)2017 4211
y Fi(+)22 b(2)g Fb(\001)f Fi(5)28 b(=)f(0)p Fh(:)716
4415 y Fk(F)-7 b(r)i(om)33 b(the)i(e)-5 b(quation)35
b(for)g(the)f(tangent,)1938 4619 y Fh(y)c Fi(=)e Fb(\000)p
Fi(1)p Fh(:)716 4841 y Fk(Thus)43 b(the)g(tangent)g(at)h
Fh(D)i Fk(me)-5 b(ets)43 b(the)g(curve)g(again)g(in)g(the)g(p)-5
b(oint)43 b Fh(E)49 b Fi(=)716 4962 y(\(0)p Fh(;)17 b
Fb(\000)p Fi(1\))p Fk(:)1480 5082 y Fi(8)p Fh(B)33 b
Fi(=)28 b Fb(\000)p Fi(4)p Fh(C)34 b Fi(=)28 b(2)p Fh(D)i
Fi(=)e Fb(\000)p Fh(E)34 b Fi(=)27 b Fh(B)5 b(:)716 5249
y Fk(In)34 b(other)h(wor)-5 b(ds,)34 b(the)h(or)-5 b(der)34
b(of)h(the)g(p)-5 b(oint)34 b Fh(B)f Fi(=)28 b(\(0)p
Fh(;)17 b Fi(1\))34 b Fk(is)g(7.)716 5388 y(Final)5 b(ly,)34
b(the)h(p)-5 b(oint)34 b Fb(\000)p Fh(B)g Fi(=)27 b(\(0)p
Fh(;)17 b Fb(\000)p Fi(1\))35 b Fk(has)f(or)-5 b(der)35
b(14.)p eop end
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TeXDict begin 5 4 bop 377 573 a Fi(9.)48 b(Sho)m(w)34
b(that)e(the)h(p)s(olynomial)1866 693 y Fh(x)1921 652
y Fc(2)1983 693 y Fi(+)22 b(1)501 868 y(is)33 b(irreducible)i(o)m(v)m
(er)e Fg(F)1351 883 y Fc(3)1391 868 y Fi(.)43 b(Is)33
b(it)g(primitiv)m(e?)501 1029 y(Find)g(the)g(group)f(on)h(the)g
(elliptic)h(curv)m(e)1752 1249 y Fh(y)1804 1208 y Fc(2)1870
1249 y Fi(=)28 b Fh(x)2029 1208 y Fc(3)2091 1249 y Fi(+)22
b Fh(x)501 1469 y Fi(o)m(v)m(er)34 b Fg(F)771 1486 y
Fc(3)806 1467 y Fa(2)845 1469 y Fi(.)501 1631 y(Sk)m(etc)m(h)h(the)e
(pro)s(of)f(that)g(the)h(addition)g(on)f(an)h(elliptic)g(curv)m(e)i(is)
e(asso)s(ciativ)m(e.)501 1793 y Fj(Answ)m(er:)538 1997
y Fk(\(a\))48 b(If)34 b(the)h(p)-5 b(olynomial)1818 2117
y Fh(p)p Fi(\()p Fh(x)p Fi(\))28 b(=)f Fh(x)2184 2076
y Fc(2)2246 2117 y Fi(+)22 b(1)716 2291 y Fk(wer)-5 b(e)30
b(not)g(irr)-5 b(e)g(ducible,)30 b(it)h(would)e(have)h(a)g(r)-5
b(o)g(ot)30 b(in)g Fg(F)2636 2306 y Fc(3)2676 2291 y
Fk(.)43 b(But)30 b(none)g(of)f Fi(0)p Fh(;)17 b Fb(\006)p
Fi(1)716 2412 y Fk(ar)-5 b(e)34 b(r)-5 b(o)g(ots)41 b
Fi(mo)s(d)5 b(3)p Fk(.)45 b(Henc)-5 b(e)34 b Fh(p)p Fi(\()p
Fh(x)p Fi(\))i Fk(is)e(irr)-5 b(e)g(ducible.)543 2574
y(\(b\))48 b(T)-7 b(o)31 b(determine)g(if)h(it)h(is)f(primitive,)g(we)f
(must)i(\014nd)e(the)i(or)-5 b(der)31 b Fh(d)h Fk(of)g
Fh(x)c Fi(mo)s(d)716 2694 y Fh(p)p Fi(\()p Fh(x)p Fi(\))p
Fk(.)45 b(We)35 b(know)f(that)1806 2814 y Fh(d)27 b Fb(j)g
Fi(3)1988 2773 y Fc(2)2050 2814 y Fb(\000)22 b Fi(1)28
b(=)f(8)p Fh(:)716 3010 y Fk(The)34 b(or)-5 b(der)35
b(of)f Fh(x)i Fk(is)e(not)h(2,)g(sinc)-5 b(e)1649 3230
y Fh(x)1704 3188 y Fc(2)1766 3230 y Fb(\000)23 b Fi(1)k
Fb(6\021)h Fi(0)g(mo)s(d)f Fh(p)p Fi(\()p Fh(x)p Fi(\))p
Fh(;)716 3470 y Fk(But)1431 3591 y Fh(x)1486 3550 y Fc(2)1553
3591 y Fb(\021)h(\000)p Fi(1)56 b(=)-17 b Fb(\))55 b
Fh(x)2109 3550 y Fc(4)2177 3591 y Fb(\021)28 b Fi(1)f(mo)s(d)h
Fh(p)p Fi(\()p Fh(x)p Fi(\))p Fh(:)716 3765 y Fk(Henc)-5
b(e)34 b Fh(x)i Fk(is)e(of)h(or)-5 b(der)34 b(4,)h(and)f(so)h(is)f(not)
h(primitive.)543 3927 y(\(c\))48 b(The)34 b(elements)g(of)h
Fg(F)1492 3944 y Fc(3)1527 3925 y Fa(2)1600 3927 y Fk(ar)-5
b(e)35 b(r)-5 b(epr)g(esente)g(d)34 b(by)h(the)g(p)-5
b(olynomials)1767 4147 y Fh(at)22 b Fi(+)g Fh(b)29 b
Fi(mo)s(d)e Fh(p)p Fi(\()p Fh(t)p Fi(\))p Fh(;)716 4367
y Fk(wher)-5 b(e)33 b Fh(a;)17 b(b)28 b Fb(2)g(f)p Fi(0)p
Fh(;)17 b Fb(\006)p Fi(1)p Fb(g)p Fk(.)44 b(\(We)33 b(have)g(change)-5
b(d)32 b(the)i(variable)f(to)g Fh(t)h Fk(to)g(avoid)716
4487 y(c)-5 b(onfusion)34 b(with)g(the)h Fh(x)p Fk(-c)-5
b(o)g(or)g(dinate.\))716 4628 y(The)34 b(homomorphism)1634
4848 y Fh(\022)d Fi(:)d Fh(x)g Fb(7!)f Fh(x)2030 4807
y Fc(2)2098 4848 y Fi(:)g Fg(F)2213 4807 y Fd(\002)2213
4881 y Fc(3)2248 4862 y Fa(2)2315 4848 y Fb(!)g Fg(F)2503
4807 y Fd(\002)2503 4881 y Fc(3)2538 4862 y Fa(2)716
5068 y Fk(has)1816 5189 y Fi(k)m(er)18 b Fh(\022)31 b
Fi(=)c Fb(f\006)p Fi(1)p Fb(g)p Fh(:)p eop end
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b(that)i(ther)-5 b(e)33 b(ar)-5 b(e)32 b(just)g(4)h(squar)-5
b(es)32 b(in)g Fg(F)2527 532 y Fd(\002)2527 605 y Fc(3)2562
587 y Fa(2)2601 573 y Fk(,)g(and)g(it)h(is)f(e)-5 b(asy)32
b(to)h(se)-5 b(e)716 693 y(that)35 b(these)g(ar)-5 b(e)1311
913 y Fi(1)p Fh(;)17 b(t)1439 872 y Fc(2)1505 913 y Fi(=)28
b Fb(\000)p Fi(1)p Fh(;)17 b Fi(\()p Fh(t)22 b Fi(+)g(1\))2059
872 y Fc(2)2126 913 y Fi(=)28 b Fb(\000)p Fh(t;)17 b
Fi(\()p Fh(t)22 b Fb(\000)h Fi(1\))2668 872 y Fc(2)2735
913 y Fi(=)k Fh(t:)716 1158 y Fk(ie)1517 1403 y Fi([0)p
Fh(;)17 b Fi(1])p Fh(;)43 b Fi([0)p Fh(;)17 b Fb(\000)p
Fi(1])p Fh(;)45 b Fi([1)p Fh(;)17 b Fi(0])p Fh(;)43 b
Fi([)p Fb(\000)p Fi(1)p Fh(;)17 b Fi(0])p Fh(:)716 1644
y Fk(We)36 b(dr)-5 b(aw)36 b(up)g(a)g(table)g(for)g(the)g(9)g(values)f
Fh(x)c Fb(2)f Fg(F)2529 1661 y Fc(3)2564 1642 y Fa(2)2639
1644 y Fk(to)-5 b(gether)36 b(with)g Fh(x)3278 1608 y
Fc(3)3341 1644 y Fi(+)23 b Fh(x)716 1764 y Fk(and)34
b(p)-5 b(ossible)34 b(values)g(for)h Fh(y)t Fk(:)1790
1976 y Fh(x)p 1999 2012 4 121 v 204 w(x)2104 1940 y Fc(3)2166
1976 y Fi(+)22 b Fh(x)127 b(y)p 1635 2015 942 4 v 1793
2099 a Fi(0)p 1999 2136 4 121 v 318 w(0)238 b(0)1793
2220 y(1)p 1999 2256 V 279 w Fb(\000)p Fi(1)168 b Fb(\006)p
Fh(t)1755 2340 y Fb(\000)p Fi(1)p 1999 2376 V 279 w(1)200
b Fb(\006)p Fi(1)1800 2461 y Fh(t)p 1999 2497 V 325 w
Fi(0)238 b(0)1716 2581 y Fh(t)22 b Fi(+)g(1)p 1999 2617
V 123 w Fb(\000)p Fh(t)h Fi(+)f(1)108 b Fb(\000)1715
2701 y Fh(t)22 b Fb(\000)h Fi(1)p 1999 2738 V 121 w Fb(\000)p
Fh(t)g Fb(\000)g Fi(1)107 b Fb(\000)1761 2822 y(\000)p
Fh(t)p 1999 2858 V 287 w Fi(0)238 b(0)1677 2942 y Fb(\000)p
Fh(t)23 b Fi(+)f(1)p 1999 2978 V 123 w Fh(t)g Fi(+)g(1)147
b Fb(\000)1676 3063 y(\000)p Fh(t)23 b Fb(\000)g Fi(1)p
1999 3099 V 121 w Fh(t)g Fb(\000)f Fi(1)146 b Fb(\000)716
3302 y Fk(Thus)32 b(ther)-5 b(e)33 b(ar)-5 b(e)32 b(8)h(p)-5
b(oints)32 b(on)g(the)h(curve)g(\(including)e(the)i(p)-5
b(oint)32 b(at)h(in\014n-)716 3422 y(ity\).)716 3564
y(A)n(lso,)h(ther)-5 b(e)35 b(ar)-5 b(e)35 b(3)f(p)-5
b(oints)35 b(of)f(or)-5 b(der)35 b(2)f(\(with)h Fh(y)c
Fi(=)c(0)p Fk(\).)716 3705 y(Ther)-5 b(e)34 b(ar)-5 b(e)35
b(3)f(ab)-5 b(elian)34 b(gr)-5 b(oups)35 b(of)f(or)-5
b(der)35 b(8:)1118 3925 y Fg(Z)p Fh(=)p Fi(\(8\))p Fh(;)45
b Fg(Z)p Fh(=)p Fi(\(4\))22 b Fb(\010)g Fg(Z)p Fh(=)p
Fi(\(2\))p Fh(;)44 b Fg(Z)p Fh(=)p Fi(\(2\))22 b Fb(\010)h
Fg(Z)p Fh(=)p Fi(\(2\))f Fb(\010)h Fg(Z)p Fh(=)p Fi(\(2\))p
Fh(:)716 4145 y Fk(These)34 b(have,)g(r)-5 b(esp)g(e)g(ctively,)34
b Fi(1)p Fh(;)17 b Fi(3)p Fh(;)g Fi(7)34 b Fk(elements)g(of)g(or)-5
b(der)35 b(2.)716 4286 y(We)g(c)-5 b(onclude)34 b(that)h(the)g(gr)-5
b(oup)35 b(on)f(the)h(curve)g(is)1791 4506 y Fg(Z)p Fh(=)p
Fi(\(4\))22 b Fb(\010)h Fg(Z)p Fh(=)p Fi(\(2\))p Fh(:)p
eop end
PStoPSsaved restore
%%Trailer

userdict /end-hook known{end-hook}if
%%EOF