虚二次域例两（-5/50）

lilianjie

% ^, w' F. e/ ^: h+ k* G6 z  KQ5:=QuadraticField(-5) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
1 g7 u' X1 c- v1 s1 }; tM:=MaximalOrder(Q5) ;
M;
* l! d# m8 {7 PNumberField(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;
IsQuadratic(Q5);
IsQuadratic(S1);
& Q) Y% E) ]3 |: E! b3 FIsQuadratic(S4);
* P; v2 P) v9 L' tIsQuadratic(S25);
IsQuadratic(S625888888);
% Y" `5 J' Y; vFactorization(w^2+5);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
: `, r8 v3 c) o2 _$ \, _8 w  V4 jFundamentalUnit(M);
: g8 x3 ~4 ?1 d& P& h, zConductor(Q5) ;

Name(M, -5);
Conductor(M);
2 p: _* a( H& o  ~: \- L# uClassGroup(Q5) ;
% H$ n+ i6 Q4 T6 V8 d, P  xClassGroup(M);
+ y4 ~: q" u" B+ `$ ]% Q: jClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
* `8 ?& L6 u7 C3 E( P
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
" ^) Z# p4 }1 p- D- ?NormEquation(Q5, -5) ;
$ c/ y' B" ?0 XNormEquation(M, -5) ;
% ^7 {4 B/ p/ b( S/ |: aQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
( P4 o5 k, @3 JUnivariate Polynomial Ring in w over Q5
* W# r: ^9 u. V1 ~* M( m4 hEquation Order of conductor 1 in Q5
5 k3 A; H, i1 NMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Order of conductor 625888888 in Q5
! Q( `( Q/ [  N) G% I8 Utrue Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
# E8 N  B" ?* j( Ltrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
- J# W/ W- t/ V2 C4 j6 P- p* D[
* ]9 s7 T' e( X" {9 t7 C    <w - Q5.1, 1>,
* {# j& i$ w: M) i5 `9 \! K# N( p    <w + Q5.1, 1>
]
-20

>> FundamentalUnit(Q5) ;
                  ^
1 f0 [' |6 O( k9 fRuntime error in 'FundamentalUnit': Field must have positive discriminant


% l. `6 o7 l3 I3 R  \! d>> FundamentalUnit(M);
3 j- ?. w: R- A9 u- I# R                  ^
$ [8 }. J# o6 ^) v- l7 ARuntime error in 'FundamentalUnit': Field must have positive discriminant
& D% E- n( x+ t
20
' l6 a7 N3 R* z5 C# @
: J' m2 ]- P8 c# K>> Name(M, -5);
9 V  n' o5 c6 k; q       ^
Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]
1 K5 V3 c0 \% V( ]& V* \5 a
& h1 K# w+ |. z6 K9 Y' m1
Abelian Group isomorphic to Z/2
; H  n7 I/ X4 @. ?( sDefined on 1 generator
Relations:
    2*$.1 = 0
) o5 k( O( t, }7 O4 YMapping from: Abelian Group isomorphic to Z/2
" K) ?! r  b3 T* `7 WDefined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
1 s$ F) M( O4 WRelations:
    2*$.1 = 0 to Set of ideals of M
2
! b% s" e: i  q* H6 ]2
Abelian Group isomorphic to Z/2
Defined on 1 generator
  ~1 h6 c' ^5 G* C9 P5 y% J) a' ^Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
" y( X+ {+ r1 |* p* }7 sDefined on 1 generator
Relations:
; Y" m. a0 A7 c/ |8 b    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
2
Abelian Group isomorphic to Z/2
: z' J( h3 B& \! X: j2 LDefined on 1 generator
+ R. @7 O9 o4 b8 i5 d9 y! bRelations:
    2*$.1 = 0
6 h( k, z. e' S. ^# ~  K8 a" CMapping from: Abelian Group isomorphic to Z/2
$ o" C) f( o4 tDefined on 1 generator
2 Z! A1 Q; \9 \' nRelations:
8 x1 K' `/ W8 q* r! l1 @1 I& c    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
" ?; `+ d& F8 J9 N& ~/ aAbelian Group isomorphic to Z/2
4 ?$ K  P& K. f1 a& ~/ f# EDefined on 1 generator
# P7 t. w8 t) S5 d) Q; P* l/ GRelations:
7 H4 J7 j- C0 f0 N( X# K+ w    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
0 I/ G* E" T7 S, p2 S$ D2 h* ADefined on 1 generator
& G- u4 ~6 l+ ?8 PRelations:
* \" C6 N4 C7 h2 x$ y& c1 B    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
( x1 m' s* q  ~' A' {inverse]
! |% J" G/ @/ E8 o0 Mfalse
; N0 L+ M5 m/ ^- ^false
==============

5 U# S" _9 n  o
5 O4 H- V, `  y/ b7 h' sQ5:=QuadraticField(-50) ;
Q5;
6 j( f* E9 a( W! p* Y. Y( U( ~2 w
5 m/ J& H/ R" @$ r: gQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
( Y2 u0 U: d* i1 Q6 ]3 YNumberField(M);
' }0 a- `8 K  g/ x! U! N# YS1:=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;
( f' A# q" t4 D! V1 sIsQuadratic(Q5);
IsQuadratic(S1);
" T0 ^0 p2 H) E( y8 Y+ d* S- vIsQuadratic(S4);
IsQuadratic(S25);
$ T" J% L2 R3 y2 QIsQuadratic(S625888888);
' }8 ?1 ^3 s7 m4 D' QFactorization(w^2+50);  
Discriminant(Q5) ;
# y! z/ n2 l7 WFundamentalUnit(Q5) ;
: ^, e' Z3 ?3 o9 t: ?, j) zFundamentalUnit(M);
. j' w: ?: d' GConductor(Q5) ;
  O- @4 f" H- p) x& M. `: V7 R
; T5 @% c* J; q: }Name(M, -50);
Conductor(M);
8 u( j" `8 j  C: f( D( c4 f1 vClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
3 e2 [, x2 m8 n5 ?6 CClassNumber(M) ;
' @8 i% H8 \2 jPicardGroup(M) ;
4 V/ r3 }( Y1 W. N. xPicardNumber(M) ;

  R: R2 N0 N/ @' ~* v6 Y8 [1 L( P/ CQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
. E" l( ]& Z! Q; h' R' yNormEquation(Q5, -50) ;
NormEquation(M, -50) ;

8 o% {( b* D2 z# h3 Y9 f4 KQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
! F( q% O" u! K. [& ^' N  h3 gMaximal Equation Order of Q5
# \- R% G2 g1 F1 v! W" mQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Order of conductor 625888888 in Q5
# S0 G0 r/ A* I& W. Ntrue Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
true Maximal Equation Order of Q5
/ S0 _' d% b' N" U- Gtrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
% J7 ^  z$ e( F1 strue Order of conductor 1 in Q5
[
: t# t8 V- l  Z+ g- V4 `    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
-8

* m9 j; p, h, v. z& @, V5 _  S>> FundamentalUnit(Q5) ;
2 J  K/ n( Q7 j/ @/ _- |1 s# b. H                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
$ ~# u" ~! h' F# c# O

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

8

>> Name(M, -50);
6 O0 i* R' Q8 p" K# p       ^
Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]
' w5 o: ?& M1 h  w
1
. O2 Z6 p$ G3 i- NAbelian Group of order 1
0 l3 ?8 q3 k9 h4 A$ f* KMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
# m6 _/ I9 N7 \) WMapping from: Abelian Group of order 1 to Set of ideals of M
: X+ v+ Y% o2 ~( m/ P' k4 S1
* @1 x% D+ f; Z0 R$ @0 \2 X6 l# Y% d1
Abelian Group of order 1
' f2 P. w3 y0 E( d8 J6 I& f0 F! lMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
; \( Z% n" p1 G: G3 b2 @4 i- M! l  ginverse]
, M& ~! d. A, Y1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
- a( b* a5 b3 q9 C  N% v* ]false
false
1 M( T& l% U; i" P

lilianjie

看看-1.-3的两种：

Q5:=QuadraticField(-1) ;
Q5;

) |" {* _+ x" L5 ^; p0 qQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
7 {. L: ], s( O0 u5 QNumberField(M);
9 h. K- t$ f0 c  U4 iS1:=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;
6 x; Q( z: W5 s- [( l& EIsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
0 v6 z" B" A6 l% ~IsQuadratic(S25);
2 q7 P% o1 ^2 o6 d% t2 t5 CIsQuadratic(S625888888);
0 v& x7 M! \0 G# ZFactorization(w^2+1);  
1 I3 t7 O6 h" TDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
! ~  q' E0 ~8 z/ _Conductor(Q5) ;

# f8 ^* f. B! |8 v# |Name(M, -1);
Conductor(M);
ClassGroup(Q5) ;
! I8 e9 u0 p2 \/ v+ f  b5 B$ z) z2 yClassGroup(M);
( ~& k% w- i5 K! g5 z1 vClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
9 V) t( X) W/ p( `% N7 EQuadraticClassGroupTwoPart(M);
NormEquation(Q5, -1) ;
NormEquation(M, -1) ;
' y2 n% |* z( z$ R5 X1 }
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
: l3 c7 B. Y) L. X. l; a2 MMaximal Equation Order of Q5
0 y" t7 @5 T. C6 U; _! X2 y1 \Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
3 U+ M6 E$ `; \" }- k* R6 ~true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
$ U$ W2 ^4 A1 L/ b6 v* P( xtrue Order of conductor 1 in Q5
) `: w8 i; D5 T7 C[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
& |& A$ _$ U2 L]
1 p* W8 E# d' x, B-4

>> FundamentalUnit(Q5) ;
                  ^
& j1 `' e) z7 c6 e, eRuntime error in 'FundamentalUnit': Field must have positive discriminant


>> FundamentalUnit(M);
                  ^
5 k! x6 O) s$ jRuntime error in 'FundamentalUnit': Field must have positive discriminant

$ Q3 _4 G$ W5 \' ~0 h: u( a4

>> Name(M, -1);
: H: p: r: O0 V6 L( v       ^
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]

) m8 Q3 ?& v* A  `1
- b/ P% x: ?7 k  {# w; I" {Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
4 G9 f9 o, }1 y' Q2 D1
Abelian Group of order 1
  k9 h! ?2 C/ H& _# h2 A2 oMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
' e* y5 R: s# v) E1 Z3 Linverse]
# B3 Q: Y- v& Q& R6 Y1 {3 S  c$ i1
! }2 ^) s$ I* E& `) eAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
" n$ e' S8 i7 v/ ?-4 given by a rule [no inverse]
% E% s0 u9 {& ^3 O; RAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
! d- b9 Z8 R  u- Y-4 given by a rule [no inverse]
, n8 v1 l* y6 h' q( M% {false
false
6 S; @( r- u. z# ]===============

Q5:=QuadraticField(-3) ;
4 g/ U' t2 `* [- |$ Q6 ^/ ?6 |* LQ5;
+ p" T: g/ K( O4 m4 N- Q& o
Q<w> :=PolynomialRing(Q5);Q;
3 ^* x8 D3 J4 G* a9 m  fEquationOrder(Q5);
, S1 [- f# ~3 R2 ?3 BM:=MaximalOrder(Q5) ;
M;
NumberField(M);
( s+ J- C4 Z! q8 D3 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;
IsQuadratic(Q5);
+ G. c$ R( ]) v& w& w. a6 A0 CIsQuadratic(S1);
! Z5 p( T' U. D% W* HIsQuadratic(S4);
' U( s3 V7 e' W0 C5 CIsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+3);  
0 g- N: B+ d0 \3 k9 b3 L6 d8 g! kDiscriminant(Q5) ;
$ o* J+ ~# b1 S) O- c: DFundamentalUnit(Q5) ;
. e: x) I0 U7 ?5 s3 ~FundamentalUnit(M);
0 u5 m7 c$ y0 c5 ^, E  D, WConductor(Q5) ;
6 }* n+ l0 i$ A+ d& I
Name(M, -3);
4 y- P7 @: X) M: q) M! DConductor(M);
ClassGroup(Q5) ;
6 p/ ~9 F7 Q- f4 B- U$ n! i; ?ClassGroup(M);
- [0 }& m' [3 P5 ^5 B8 f+ wClassNumber(Q5) ;
ClassNumber(M) ;
: k" S/ m2 n1 |! e1 t! U8 o0 ]PicardGroup(M) ;
3 q- Y7 b0 D" c1 [# {PicardNumber(M) ;
) f) }! U0 i, B
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
# S( W0 r# S6 u! f0 ~/ t6 ~! WNormEquation(Q5, -3) ;
NormEquation(M, -3) ;

6 g  h5 ?- x$ {; F1 A( o" AQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
1 W' G  R5 u( w6 w7 ]Maximal Order of Q5
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Order of conductor 625888888 in Q5
7 Z- q* j5 y* s' Etrue Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
* G' f8 {0 O9 z2 I# f4 z4 atrue Maximal Order of Q5
true Order of conductor 16 in Q5
' b* o5 c7 ~% e- b3 G: etrue Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
[
    <w - Q5.1, 1>,
& a3 N6 P2 z; L( O3 ?5 R    <w + Q5.1, 1>
]
. Y9 q; @# Q/ b) V-3
1 `" x- Q8 k' x' k2 Z
8 D  {7 W( H3 E2 `4 k; r" h5 b- _>> FundamentalUnit(Q5) ;
                  ^
& |# s# x0 m# d3 i9 r( d0 TRuntime error in 'FundamentalUnit': Field must have positive discriminant


% q) S5 X+ @1 w" ^2 B>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

% J! q) I8 H9 R3 E3
% v$ j1 R( U2 X/ a# S! ^% X9 B
>> Name(M, -3);
       ^
( K1 V2 l8 N+ e" z% xRuntime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]

1
- F- Y" ]* v/ r& O, n* LAbelian Group of order 1
+ q  r/ g, @6 i( dMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
% d2 f# c, {" ~) \) y7 qMapping from: Abelian Group of order 1 to Set of ideals of M
; l- n2 \5 }9 T1
1
6 o& T7 Y7 `( K; BAbelian Group of order 1
& ]" H! L* g; m' T3 |$ AMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
# g3 e. A3 h( Q' B, D1 minverse]
+ i5 A3 Q) ]/ X' C1
7 L+ V/ z2 `- T1 d+ V) AAbelian Group of order 1
* G+ ?0 P; J' @5 `$ l3 O" CMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
) K* Z: @+ y5 e/ Z% }3 h-3 given by a rule [no inverse]
: ~0 P% T4 B  TAbelian Group of order 1
  K; [1 R9 S- O  t1 `" ]7 DMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule [no inverse]
false
false

孤寂冷逍遥

lilianjie

Dirichlet character
Dirichlet class number formula

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
+ y0 i# H4 z9 D( f/ Y
-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

6 ^1 d( z! y5 P5 F-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
3 U; _) w) V' j" l$ `0 G, Y. |! H/ `h=-6/（2*3）*Σ[1*1+（2*（-1）]=1

-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，
7 R$ G, Q" e* B+ U  x
. H* U. |% A; S! m( ^+ a

h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2

& j# x- O$ T+ d  ^5 S
& Z4 @+ j$ p2 [9 Z  B* v
1 W9 x5 `% }$ K7 k# i-50时  个单位根                          N=200
* F2 i8 A$ q0 n- f4 A2 |9 Q

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

8 x) T# @# `/ L* l* m; jF := QuadraticField(NextPrime(5));

0 j6 {: h9 a1 _1 Z& H4 YKK := QuadraticField(7);KK;
+ P) a' h& W$ h/ O+ n0 V* ]K:=MaximalOrder(KK);
. {: A' s' {$ z# `9 J5 X# \Conductor(KK);
ClassGroup(KK) ;
QuadraticClassGroupTwoPart(KK) ;
, g+ u2 E1 V( Z( vNormEquation(F, 7);
A:=K!7;A;
B:=K!14;B;
Discriminant(KK)
& I* D+ u& a( U5 i1 i
& m( H  e1 U9 n0 W4 ?Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
6 P1 O5 b/ d$ S$ t28
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of K
Abelian Group isomorphic to Z/2
. }( S6 N+ ]& EDefined on 1 generator
0 u, _) D4 D* _& p2 DRelations:
0 f1 o# n% K& S0 N2 w5 k9 p    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
7 I, z7 X$ K6 |8 RRelations:
9 t/ [4 T/ e! T" S) B    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
inverse]
false
7
0 d2 e9 }. g/ Z' f+ i/ A14
; i2 l+ x- X4 w, R28

lilianjie

# X9 V' t) Q' v9 U8 B: j, L9 ~0 {8 t8 m0 x

* S, w* }( Z7 ]) O$ a& R; p
5 t3 s( \+ h- L' l1 x0 h



分圆域：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);
C:=CyclotomicField(6);C;
CyclotomicPolynomial(6);
CC:=CyclotomicField(7);CC;
! @5 C1 k. @& _( SCyclotomicPolynomial(7);
$ s+ L3 P  }9 c8 _MinimalField(CC!7) ;
7 \( Z% ?8 K1 P2 tMinimalField(CC!8) ;
MinimalField(CC!9) ;
( g* q1 W8 z' j0 R9 |5 R6 S, ^MinimalCyclotomicField(CC!7) ;
RootOfUnity(11);RootOfUnity(111);
* b( ~8 w. k0 o  h7 l* OMinimise(CC!123);
Conductor(CC) ;
CyclotomicOrder(CC) ;
& ]9 m% Y+ i5 M; _* }# I
CyclotomicAutomorphismGroup(CC) ;

Cyclotomic Field of order 5 and degree 4
6 |! Q8 |; W) g( w$.1^4 + $.1^3 + $.1^2 + $.1 + 1
Cyclotomic Field of order 6 and degree 2
9 x. f) C8 Y8 j, l  ~* K' Y( w4 {1 `$.1^2 - $.1 + 1
Cyclotomic Field of order 7 and degree 6
& k; v3 \: b4 J% {- I4 I( A$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
Rational Field
9 t8 A3 \) H3 n$ Q9 X1 WRational Field
Rational Field
. i+ _, z& R) G% F1 t  X# c+ dzeta_11
. E# G  P: s. c  U, ]zeta_111
- C; ]- O1 I7 n! G- q8 @  }123
. J, K2 h/ f$ |- Q) U7
7
5 w! M( n1 D( N' @Permutation group acting on a set of cardinality 6
. j9 E/ }* o" t$ P/ \7 j- aOrder = 6 = 2 * 3
1 b+ e6 Q+ y! @* A0 ]$ e    (1, 2)(3, 5)(4, 6)
    (1, 3, 6, 2, 5, 4)
& `9 ~  @8 P7 y/ c5 x$ \Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
# I2 ^2 I3 l+ g! n9 R" tCC
Composition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
& m9 f+ c3 R+ P  UDegree 6, Order 2 * 3 and
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
; q" f& u0 v  C2 S% bCC

lilianjie

0 _% s7 ?% x' U0 S' @

lilianjie 发表于 2012-1-9 20:44
分圆域：
C:=CyclotomicField(5);C;
& B0 a  @4 j2 E, X6 Y8 r2 t. D7 ]CyclotomicPolynomial(5);

7 ]& I3 @$ ^4 y* x9 W- E
分圆域：
分圆域：123
* z8 T  x; _7 n( ]
R.<x> = Q[]
( x& H* q* k+ d7 C5 w  V3 fF8 = factor(x^8 - 1)
F8
) `! a: G3 m8 l0 d: Y; p1 e6 y  T
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

) B5 R4 y# B$ `+ P" n2 W% Q% k' XQ<x> := QuadraticField(8);Q;
# C! `/ e, N. A5 f: j4 K2 yC:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;

& I( v+ @1 R, b" y% tF := QuadraticField(8);
F;
8 T4 O" Z6 V# u( U8 s% c' n& ED:=Factorization(FF) ;D;
" d! h% @7 q: P  O% p* S* eQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
' }% w% x6 ]7 V; B8 G2 X3 d1 B$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
1 c0 v8 f0 G# |8 @[
  R1 _8 P# y! F0 p7 e, S! P3 s    <$.1^4 + 1, 1>
]
) t: |, K, |5 @) r7 x- x5 o
R.<x> = QQ[]
2 G) J1 F/ S7 M5 A4 N% qF6 = factor(x^6 - 1)
! L/ L, ]9 S' D+ F) k! t$ Q' DF6

" S( @- l0 t0 j(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
: v$ K4 N5 X2 |- `( L& q# N
Q<x> := QuadraticField(6);Q;
! q! z1 }% h% j' TC:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
& H2 F* t/ c6 O3 _F;
4 v( l4 [$ [; cD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
& H8 L0 b4 P9 j. D( @9 X# K$.1^2 - $.1 + 1
4 [8 ?/ E$ K6 t: E: D3 t- N) B6 n# f  JQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
/ A7 O( Z% z; T, {4 m    <$.1^2 - $.1 + 1, 1>
]
! \, r7 k8 f* B0 p0 B9 I2 m+ P
R.<x> = QQ[]
( A2 N9 B! o$ q% tF5 = factor(x^10 - 1)
3 G3 P5 d: h* _; {, iF5
2 s0 d$ x/ h) G- I0 @& P% _. m(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
% r4 E/ S2 M* a+ h1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

9 [4 `2 o/ U0 B: VQ<x> := QuadraticField(10);Q;
( Q* {) T) a0 f9 VC:=CyclotomicField(10);C;
3 [0 E* ^+ M/ r' O+ aFF:=CyclotomicPolynomial(10);FF;

F := QuadraticField(10);
F;
- f* o2 O7 H9 b( a% dD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
2 F' q; s3 v* B2 o$.1^4 - $.1^3 + $.1^2 - $.1 + 1
' {. A* U' k# n1 D% f- _5 a" z" GQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
3 r: G3 ?* l3 W]

lilianjie1

分圆域：123