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标题: 虚二次域例两（-5/50） [打印本页]

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作者: lilianjie    时间: 2012-1-4 17:41
标题: 虚二次域例两（-5/50）

% u9 X! u1 J9 I5 c; z+ a1 K
Q5:=QuadraticField(-5) ;
% D3 _4 B1 I7 U: W+ E3 ^Q5;
8 A+ f! E: j. O
Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
) m! e: |0 ?+ p& DM:=MaximalOrder(Q5) ;
7 S' L2 n! ]4 E) o' H" ^& uM;
NumberField(M);
3 R% [% t- Q7 A1 U. W% o. C9 ]) o* LS1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888;
IsQuadratic(Q5);
9 @6 t* S6 E/ O( V5 DIsQuadratic(S1);
IsQuadratic(S4);
1 @% I( S+ ]7 z8 t# TIsQuadratic(S25);
8 o" c4 t8 J* Z5 ~3 b  H% dIsQuadratic(S625888888);
4 G8 r. Q4 r2 g5 b5 qFactorization(w^2+5);  
6 ^" d4 y0 ]% i4 J: K! ^' {) y; U1 _Discriminant(Q5) ;
- ^. m. q; B' oFundamentalUnit(Q5) ;
4 J0 [! O8 _/ p2 E# I  L; c7 v- NFundamentalUnit(M);
2 D0 ]8 d9 W4 V4 O! r6 ZConductor(Q5) ;
( g$ E- }: _: J! \' U. e  E% T7 h
% r  a7 D' n8 z1 D& y" E- j7 {Name(M, -5);
Conductor(M);
% j6 F; v6 o5 Y- ?% G2 ~8 BClassGroup(Q5) ;
7 T6 M7 ]8 h2 K6 i: x& m$ d% cClassGroup(M);
ClassNumber(Q5) ;
" i' z" l0 M' b2 }. s, e6 m) T2 mClassNumber(M) ;
' C( i& v3 @  k  h+ s8 FPicardGroup(M) ;
* P* P5 E; H' u- c# mPicardNumber(M) ;

' e  d8 v2 Q* k2 Q* v8 s. SQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -5) ;
* k- W$ J6 `, i0 K: M- u5 h0 dNormEquation(M, -5) ;
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
/ t5 p& k0 ?' V4 x/ @4 ~* P4 [/ L( ?0 gEquation Order of conductor 1 in Q5
& x& [& R* ]" A: Z1 mMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
; ~+ p2 s* {& I8 F  p  a+ f+ M# U6 Htrue Maximal Equation Order of Q5
& k  e: o. I; ]% r. q7 r* Ptrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
6 q, r. v6 N- D( y7 y2 V    <w - Q5.1, 1>,
    <w + Q5.1, 1>
4 B- {% c8 B0 d! o7 c- |3 G# d8 E]
-20
4 O3 D( S( s" n0 n6 O4 S
>> FundamentalUnit(Q5) ;
                  ^
) k& Z; Z( C) ]Runtime error in 'FundamentalUnit': Field must have positive discriminant


# Q8 S- G' n/ ^8 |. C0 h7 E>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

) T  |6 M9 }2 P2 b$ s20

>> Name(M, -5);
% t* m$ I$ _' w3 b: ?3 i       ^
8 u6 E2 Y, l* r' m! X4 H$ eRuntime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]
$ \& @# G9 b( U  a
; c# A2 f0 S9 E! C1
+ d' ?1 C* W2 Q5 F0 w2 VAbelian Group isomorphic to Z/2
3 \) k; D, e) S" [* J: ODefined on 1 generator
" ]9 G' l) f% H$ A  YRelations:
    2*$.1 = 0
7 s5 [4 n1 v% \# M/ }Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
+ p) j, F) ?& F  p* qRelations:
    2*$.1 = 0 to Set of ideals of M
Abelian Group isomorphic to Z/2
7 H* Z* h# @; ]% _Defined on 1 generator
3 B  W+ i/ q$ {4 G) @4 k- C" J( \1 KRelations:
+ I) J( |" S4 D1 q( B; c/ c    2*$.1 = 0
# i& j, j3 u; h3 l% u6 B" [Mapping from: Abelian Group isomorphic to Z/2
3 `% l; Q( i# rDefined on 1 generator
Relations:
4 y) E- x) H' U; N    2*$.1 = 0 to Set of ideals of M
0 L; ?2 E- j. e) ~1 \4 ^* ?2
2
  m  ^$ q5 [# {; R/ y2 tAbelian Group isomorphic to Z/2
# ~0 |% L, B7 m  YDefined on 1 generator
Relations:
4 c/ X" n' M6 @1 w+ d. C0 @    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
% E! U) K% k( p9 O$ i* q8 aDefined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
6 }; c* R" e7 G3 U  l* G& k/ }- K2
Abelian Group isomorphic to Z/2
( Y  N) I- e: f) jDefined on 1 generator
Relations:
3 s1 r* t' G' L+ z    2*$.1 = 0
! P( l% T* I9 v; h2 gMapping from: Abelian Group isomorphic to Z/2
* E- p& l8 g4 `+ lDefined on 1 generator
) A  v& j; z* X0 A* j  }6 ~* bRelations:
4 x5 T# Y. j4 q5 P    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
9 ~  Z: Z% B, `inverse]
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
1 _+ \- ^8 W7 F    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
9 q4 L3 k9 G+ D9 @' uRelations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
" e# M8 s4 N0 ]false
false
==============


8 L+ Z/ e; z% U, F8 G7 Z- eQ5:=QuadraticField(-50) ;
- T! F  L9 z  D) b9 v4 G5 hQ5;
9 T7 k' ^8 I" W6 g: \
Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
* f/ m  i. G, [: ?M;
: V( y+ i( J; @7 r/ ONumberField(M);
S1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888;
. t) u- o: b4 {+ kIsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+50);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
% m$ K9 G/ S$ Y5 t* NConductor(Q5) ;

/ m. E( `* o4 Z1 z! {Name(M, -50);
& X9 H0 D% S8 i* D/ E( v. eConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
5 A5 M' L# U0 {7 F, p5 W) s# wClassNumber(Q5) ;
5 J2 h5 O7 _+ O7 \9 m' yClassNumber(M) ;
/ x: V: r) P9 m- PPicardGroup(M) ;
$ h' {! ]4 D$ h& \7 x/ ~: LPicardNumber(M) ;
( n) D! l2 o6 ^, p, w, ^
QuadraticClassGroupTwoPart(Q5);
6 G1 ^! g! V! E3 N/ TQuadraticClassGroupTwoPart(M);
  q# ^: m$ X! _, qNormEquation(Q5, -50) ;
( \: d9 Y- J* l7 o, QNormEquation(M, -50) ;

Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
' ~. _. n% P, w+ yUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
) B7 a! P4 B2 YQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
4 q, }- o' B. ytrue Maximal Equation Order of Q5
6 _. _# N# I/ J+ R! Ctrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
    <w - 5*Q5.1, 1>,
4 }' v- Z/ k0 {% T# l- h. o    <w + 5*Q5.1, 1>
]
-8

/ i- L( X/ O/ O>> FundamentalUnit(Q5) ;
5 H1 v2 Y( X7 a                  ^
# {% a; [( d9 @/ G) d6 JRuntime error in 'FundamentalUnit': Field must have positive discriminant
4 R2 I$ e% h1 C' V& [0 G; u2 ~
4 N+ W' C' E+ O# i
# @; R* S# E" i% P  G' [6 }; _: u: U>> FundamentalUnit(M);
& W2 _6 m/ @2 T) u& s: K  o                  ^
8 M; c4 W* x6 WRuntime error in 'FundamentalUnit': Field must have positive discriminant

7 {2 P7 L' N% i, R1 o+ a8

>> Name(M, -50);
       ^
% T7 ?7 O4 w  iRuntime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]

1
' i: v- X* Q5 |  n( r9 xAbelian Group of order 1
. H6 X; q, Z5 E* sMapping from: Abelian Group of order 1 to Set of ideals of M
) v) n( k8 C6 AAbelian Group of order 1
" i2 y7 C% D% f' OMapping from: Abelian Group of order 1 to Set of ideals of M
1
2 l8 G( N8 K1 Q' r1
/ E8 j( n7 C+ B4 |$ _Abelian Group of order 1
8 ?( V$ y) H0 ]Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
. {. \  i. T6 G5 yinverse]
1
# P! d% p% p, k# R" `Abelian Group of order 1
3 Z5 s8 g0 q* K& V* I% jMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
* _2 o4 ?( j7 Z- TAbelian Group of order 1
8 K- S" @5 ^' {% eMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
& F! S" l& c) C" K1 Q6 M-8 given by a rule [no inverse]
7 _5 k# k6 L8 }' @7 X2 Pfalse
2 W4 V% G# t1 b4 ]' Y" R# J6 @false
: W0 g; `; q. {% e- j9 K$ {

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作者: lilianjie    时间: 2012-1-4 17:51
看看-1.-3的两种：
0 `7 O  q' U( f" }( q/ i% n1 q
Q5:=QuadraticField(-1) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
% E! w" z, y9 }M;
NumberField(M);
6 T: }* U) j! HS1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888;
IsQuadratic(Q5);
" x: y1 X/ G# d3 QIsQuadratic(S1);
IsQuadratic(S4);
+ i2 J1 O& x$ V: H, Z  MIsQuadratic(S25);
5 ~) E! O7 m) t% RIsQuadratic(S625888888);
6 D7 r% J, n- V9 l8 R0 bFactorization(w^2+1);  
" @  o" Y! F! h: y+ HDiscriminant(Q5) ;
- i5 u0 a& W4 u# ~( oFundamentalUnit(Q5) ;
FundamentalUnit(M);
% F5 r' b) Q( |* mConductor(Q5) ;
8 q5 d  p/ {) I2 _1 A3 ?4 U: d) `
Name(M, -1);
4 M, r% u0 _6 h  z: ]Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
1 Y( j/ Y. x( a- S( }% u$ i* ]PicardGroup(M) ;
+ J8 e" Q- y3 y4 ^; HPicardNumber(M) ;
, U7 w* b) \# C7 v
QuadraticClassGroupTwoPart(Q5);
( W# _+ `6 Q0 |QuadraticClassGroupTwoPart(M);
5 V) c9 T: y& s3 ?& b: S  lNormEquation(Q5, -1) ;
NormEquation(M, -1) ;

- J5 k% l# Y' @Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
' K. N- R/ N& mMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
1 N9 X$ T  v) a' O. TOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
- ^6 N9 }2 Q$ O& c5 F) h  U7 ?& y-4

>> FundamentalUnit(Q5) ;
+ p* c8 W  e6 q/ v                  ^
2 P" S. t% T" t& a/ s4 LRuntime error in 'FundamentalUnit': Field must have positive discriminant
2 A& T, M8 F0 R& V

>> FundamentalUnit(M);
; \, s0 l- |; `9 j/ M# Z. ]; ]                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

0 j/ ?; f% ]; v0 [" P, h4

; k# ^! Q( j5 O4 y: q>> Name(M, -1);
       ^
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]

1
  W$ n; m" t% |Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
( C' n( t; a" l$ i. l* d9 s; jAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
" _  y6 x  p9 x2 q$ P2 v; b1
1
Abelian Group of order 1
: q$ x/ i; s  c' ^/ y% |- g4 NMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
' X0 u" q3 `8 B4 Y1 ]% G1
, C/ g% G6 a- ~( Z- G* NAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
3 O. O% u( |, m; X, I5 N-4 given by a rule [no inverse]
Abelian Group of order 1
  b  O8 o- L4 T( `/ NMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
- p5 ?3 n* `& y: X-4 given by a rule [no inverse]
false
$ D! i) M4 {2 ^0 H# `! m: o# a4 Mfalse
===============

Q5:=QuadraticField(-3) ;
Q5;
3 l( }, \7 D- y, @
Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
; V2 h3 n- h& L4 m; [* Y1 e' @M:=MaximalOrder(Q5) ;
" ^( [- p& r' D! l  ?M;
NumberField(M);
  L6 s8 Y: W+ p* ZS1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888;
IsQuadratic(Q5);
' R5 g# k" `& B5 SIsQuadratic(S1);
. I, z6 `6 T1 L: C1 @IsQuadratic(S4);
6 ~' A6 M7 B4 p8 Y7 s1 \. mIsQuadratic(S25);
IsQuadratic(S625888888);
& p8 O4 X9 x6 ]; ZFactorization(w^2+3);  
Discriminant(Q5) ;
! ?0 Z9 `1 ]* ^) PFundamentalUnit(Q5) ;
FundamentalUnit(M);
( O& a/ G+ J3 S( o" [* f! ^Conductor(Q5) ;

2 H. B, w$ z: jName(M, -3);
. d7 Z, p, k2 d+ {! s# VConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
. [# C8 F% ~  I4 m5 O0 CPicardGroup(M) ;
6 R* h8 v7 r% d4 ]2 I% z; UPicardNumber(M) ;

) u* e6 p' |" P! iQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -3) ;
NormEquation(M, -3) ;
- X+ ]( b! D1 i' X% \9 t: ^( p
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
5 K9 s  L! k' J) S6 CUnivariate Polynomial Ring in w over Q5
& G# S' U" Y4 Q8 yEquation Order of conductor 2 in Q5
Maximal Order of Q5
& _( ^( p7 e# r2 P) vQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
4 o! _% B; J1 G: x6 P3 e+ J, ~true Maximal Order of Q5
true Order of conductor 16 in Q5
2 X/ {" k! T5 p, e& C' M) Strue Order of conductor 625 in Q5
5 O5 \* M) H5 X9 w9 A6 }true Order of conductor 391736900121876544 in Q5
" b& d# U" i' Q! Y[
    <w - Q5.1, 1>,
  p7 [+ C7 Y" {& m$ r3 S: y/ n    <w + Q5.1, 1>
]
-3
2 [1 p$ }" g* O' h# q) |
>> FundamentalUnit(Q5) ;
! Y4 U& K; q0 p/ H/ c7 l                  ^
8 G3 ~" l( s% H6 Y, Y0 cRuntime error in 'FundamentalUnit': Field must have positive discriminant


>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

3

>> Name(M, -3);
! r# o! |8 j; `: d       ^
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]

2 Q, l0 Q( u  S1
Abelian Group of order 1
) P. @1 k$ M9 r4 b" TMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
3 R6 r0 {5 z2 E  MMapping from: Abelian Group of order 1 to Set of ideals of M
" z* B& o' \% m" g# x2 n1
1
, n& |5 x2 e2 g) U5 V( P, x8 mAbelian Group of order 1
7 f, t! L* l4 A" }& OMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
* v& K1 F( Z/ Y1 G8 z1
. K2 {. t$ N& b. U$ @Abelian Group of order 1
7 ~2 r; e& i; B  JMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule [no inverse]
Abelian Group of order 1
+ q, n# \; [* K+ bMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule [no inverse]
false
- F( E1 E+ u4 C* g( K3 f0 ], G0 hfalse

---

作者: 孤寂冷逍遥    时间: 2012-1-5 08:36

---

作者: lilianjie    时间: 2012-1-5 13:02

9 G: o; H/ ]1 f: o6 l; u
Dirichlet character
Dirichlet class number formula
; C  }: f8 I* y) J4 P
虚二次域复点1，实点为0，实二次域复点0，实点为2，实二次域只有两单位根
( T( S- Q+ ^, ?% D
5 Q" C- z. O! l, p8 F$ D-1时，4个单位根1，-1, i, -i,w=4,            N=4，互素（1，3），    (Z/4Z)*------->C*      χ（1mod4）=1，χ（3mod4）=-1，h=4/（2*4）*Σ[1*1+（3*（-1）]=1

; }5 [" y8 [' U-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
: P% D1 K% W% q0 S& yh=-6/（2*3）*Σ[1*1+（2*（-1）]=1
1 j1 N# e  W9 `% D$ M2 j
-5时  2个单位根                              N=20   N=3  互素（1，3，7,9,11,13,17,19），    (Z/5Z)*------->C*         χ（1mod20）=1，χ（3mod20）=-1， χ（7mod20）=1， χ（9mod20）=1，χ（11mod20）=1，χ（13mod20）=1，χ（17mod20）=1，χ（19mod20）=1，


+ s& r0 ?9 u3 h7 K
h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2


. s1 g5 p' m& i+ ~! `3 R# V1 Y
-50时  个单位根                          N=200

---

作者: lilianjie1    时间: 2012-1-5 13:51
Dirichlet character

21.JPG (79.18 KB, 下载次数: 266)

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作者: lilianjie1    时间: 2012-1-5 20:37
pell   equation

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作者: lilianjie    时间: 2012-1-9 20:28


F := QuadraticField(NextPrime(5));
" b% [1 b9 L4 J% j: z$ P8 c
2 E5 y4 O5 D! a- S6 [) Q- f3 IKK := QuadraticField(7);KK;
$ [# I3 f# }4 h* Z. pK:=MaximalOrder(KK);
Conductor(KK);
* X  `2 R, J! b  Y/ Z4 Y& |ClassGroup(KK) ;
QuadraticClassGroupTwoPart(KK) ;
NormEquation(F, 7);
A:=K!7;A;
B:=K!14;B;
Discriminant(KK)
3 T2 E' r( U2 u6 Z, \( e/ l" `. h
; F: J9 _; S7 r" {  A: y$ HQuadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
28
' R' i' |+ P8 p5 n1 W. g% R* I% p1 \Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of K
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
" }( [4 q0 s0 P7 ~9 J    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
* e/ j& ^& g/ ?4 binverse]
false
7
14
28

---

作者: lilianjie    时间: 2012-1-9 20:44


9 a# A* w8 t+ J; x. W


& M- F1 j3 S: m# s
" R% D' W8 P; w. e/ H
) a; L& V% c- [) k
9 ^1 m: X/ y1 ?5 H$ N分圆域：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);
C:=CyclotomicField(6);C;
CyclotomicPolynomial(6);
& O% R) u( b3 `! A; D, @; O( cCC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
6 g8 u, s+ q1 x/ M: k# }( DMinimalField(CC!7) ;
( i; h: U; P& s% i/ z, SMinimalField(CC!8) ;
MinimalField(CC!9) ;
MinimalCyclotomicField(CC!7) ;
. p$ h; U7 M; M! f( s/ t1 [RootOfUnity(11);RootOfUnity(111);
Minimise(CC!123);
2 |8 Z9 d, `$ G2 ZConductor(CC) ;
6 q# U2 ?/ j8 M7 RCyclotomicOrder(CC) ;
3 a2 @6 f  R% ~' E3 E
CyclotomicAutomorphismGroup(CC) ;
* }" N) b- _' Y
Cyclotomic Field of order 5 and degree 4
1 g4 c5 O( P1 f) }; m; u$.1^4 + $.1^3 + $.1^2 + $.1 + 1
Cyclotomic Field of order 6 and degree 2
: C( G& R& v6 q* z5 h) I$ y/ D$.1^2 - $.1 + 1
Cyclotomic Field of order 7 and degree 6
" H' i/ ]6 r, l  q" v$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
' Y2 c1 l! P3 U1 V. mRational Field
/ g& W6 J2 z2 s3 JRational Field
# p2 c4 w0 U" S( l( r; A( C' l% KRational Field
, M( i5 P3 ^: Q8 E* ^# K. _Rational Field
' a& w; L+ L# Z6 {zeta_11
zeta_111
4 T+ ~0 v, W  L8 ^6 L6 h123
* N+ K; [7 [' H* h# h/ u7
7
8 p3 p( h! F2 z9 b( H9 C, yPermutation group acting on a set of cardinality 6
Order = 6 = 2 * 3
    (1, 2)(3, 5)(4, 6)
    (1, 3, 6, 2, 5, 4)
  \! n5 F% H8 _/ _( PMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC
$ Y( E; W3 n' d- PComposition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
Degree 6, Order 2 * 3 and
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
, S, R, ?% i+ _4 L3 OCC

---

作者: lilianjie    时间: 2012-1-10 11:34

. [$ M& Z  U' {0 h! B

lilianjie 发表于 2012-1-9 20:44
分圆域：
7 ^0 F! I5 m1 q7 E$ Q* o: GC:=CyclotomicField(5);C;
5 v9 `* ^( _5 qCyclotomicPolynomial(5);

3 n" U* c9 }# K/ E
: V' c2 Q, W% j" L( k0 E% A分圆域：
$ ?) h# i" O, g9 d$ W分圆域：123
6 ]1 t7 ~6 Y8 B7 `
R.<x> = Q[]
+ N$ f2 q. f2 w4 V1 \4 ~F8 = factor(x^8 - 1)
) J2 \! w5 t- V+ R$ z3 SF8
. s6 d/ L! K+ j8 }* A, B
1 Y6 M' ~  j1 N(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
$ j0 K3 u* Q2 O
4 y# E; I: f6 e- h2 V/ VQ<x> := QuadraticField(8);Q;
3 }( O# D6 c; |7 h0 @. M) TC:=CyclotomicField(8);C;
. Z) T9 J. C' d" w/ z/ k" n8 M  L7 eFF:=CyclotomicPolynomial(8);FF;

F := QuadraticField(8);
F;
D:=Factorization(FF) ;D;
: ~( C: S2 }& e3 z: O/ d% gQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
9 {2 n  q1 _5 y/ D7 k+ aCyclotomic Field of order 8 and degree 4
$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
* g2 L' d( ?" ^: T; ~7 A+ p[
    <$.1^4 + 1, 1>
$ Z' y5 e5 X1 y8 _2 }4 |]
# \  u( n! |) ]
R.<x> = QQ[]
F6 = factor(x^6 - 1)
$ h' R% y' d2 E% D" {5 S3 @5 FF6
1 X0 V& g1 g$ l% A4 j+ `9 @
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

Q<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
' L8 P* i: f, g8 F+ VFF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
F;
D:=Factorization(FF) ;D;
( ]7 D  d  U1 g4 `Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
/ m9 \( ~- w. l7 b" h% n3 _$ X$.1^2 - $.1 + 1
: I) C2 _# n4 z) Q/ g. Z( T: Z2 AQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
    <$.1^2 - $.1 + 1, 1>
]

5 U: y/ p+ d  [3 LR.<x> = QQ[]
" q/ ^! W  x) jF5 = factor(x^10 - 1)
F5
6 _, B6 ?8 z, q! `(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
5 T6 M/ i; p6 R- A5 @# ]( N, j# M1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

% g% B# z/ ~3 I; X2 ^+ iQ<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
, ?% n0 h+ U# l9 Z7 CFF:=CyclotomicPolynomial(10);FF;

F := QuadraticField(10);
. k9 y( [0 n, y; v" jF;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
9 H; W) x6 y3 {4 ^% WCyclotomic Field of order 10 and degree 4
$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
9 i& R% |- y. B) ?; V# y4 }    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

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作者: lilianjie1    时间: 2012-1-10 11:55
分圆域：123

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