虚二次域例两（-5/50）

lilianjie

; e. A( \) w6 @4 h) D, v
! ]  E7 P; t/ V( Q7 |  o. ~: H( m0 UQ5:=QuadraticField(-5) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
/ }& @2 _$ Y( SM:=MaximalOrder(Q5) ;
- [" z4 E1 J5 z* S" G2 J+ hM;
NumberField(M);
$ x  n2 q, ]- f, Y/ Z: j: d8 rS1:=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);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+5);  
( g% u4 X% A- U0 B: c* l( lDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
- l; h; _) J2 b9 E# D, ^/ t- OFundamentalUnit(M);
Conductor(Q5) ;

, J$ A- Z; {" C) T: bName(M, -5);
% H8 d7 n: X& Y/ R' i+ @! p  hConductor(M);
* O  ^3 R! u" h3 t& V3 mClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -5) ;
9 D1 `$ m+ Z! WNormEquation(M, -5) ;
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
3 F! Z: A. V9 Q( ?. TUnivariate Polynomial Ring in w over Q5
. B& D* g( x& j3 Y9 @Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
4 C3 S  b  g. N7 z* h$ xOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
! z8 D% A+ c# X# B& F- ztrue Order of conductor 1 in Q5
* T' s3 I5 l! z. G. [[
1 ~9 h" i0 r  d+ R$ D    <w - Q5.1, 1>,
& m* _0 T, E; s8 B# A: @$ d    <w + Q5.1, 1>
]
% }$ r# e* Y+ Z& Z( W-20
. H4 F* h% s: K2 S( U+ n  v
>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant


! j% I( I) N6 ]' A. H, z3 K& D>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
! M8 a+ Q, ~' S7 m% i$ d
20
6 g9 K2 {3 O! u
4 z" r/ W& ]; w5 c( r>> Name(M, -5);
       ^
# M. {" q& z/ G& ZRuntime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]

/ C* A9 Z0 I/ B9 V& q6 a1 Y1
5 I1 w% T, v3 lAbelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
/ y/ X6 Y1 z- z2 p) O% l! Y0 b, oMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
" b8 n* P* T" P7 z    2*$.1 = 0 to Set of ideals of M
: y9 Z+ C8 N6 e* o! ]Abelian Group isomorphic to Z/2
2 U" Q; Q2 ?( ~- d: iDefined on 1 generator
Relations:
    2*$.1 = 0
% k5 d1 a7 g. B6 vMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
% W8 d4 e3 Y1 \! O! t    2*$.1 = 0 to Set of ideals of M
& t# u8 K8 `) u1 Q; Z0 I2 {# Y2
6 b4 X7 B% l# X3 R% [2
Abelian Group isomorphic to Z/2
6 R( o1 m8 y3 o6 M, mDefined on 1 generator
  n; s& S" w/ F5 O/ Y) B5 ARelations:
! [' u- U. C! y3 \    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
, X3 i: g9 E6 a  i8 ^Defined on 1 generator
Relations:
& M5 d4 a- ]$ V. a) I1 T& t    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
: d: K9 _& f6 ?  a2
Abelian Group isomorphic to Z/2
% ^& Z. X$ u0 {0 R. tDefined on 1 generator
) p( E$ b3 ~% f% qRelations:
8 l/ K) Q$ \( a* U1 O* `    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
1 O8 u% u. [6 l4 x1 F8 IRelations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
. B# u4 C$ Z  x  B+ c" h$ x, ~inverse]
; X' [: Q9 Z6 p( m; O; J( p+ ~Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
5 `* C- y; Q- f- A    2*$.1 = 0
) N+ s: K7 D/ E& e2 m5 N; BMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
% L: t  n6 K/ P$ TRelations:
5 `( Y, w2 s2 @- q9 _) e& x/ }    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
+ g: W  _; Y# w6 x5 }' C' N5 y  {1 Kinverse]
false
false
==============


Q5:=QuadraticField(-50) ;
8 Y. Q, ?) \3 r: w5 @9 S! {Q5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
+ }0 l" x- v& w5 N6 ENumberField(M);
8 v8 q) a2 J2 u' R, f: 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;
IsQuadratic(Q5);
IsQuadratic(S1);
* }: g4 @3 D+ z: j5 k! \0 u6 ]  i6 IIsQuadratic(S4);
& @2 e9 l: k7 u6 h# [- eIsQuadratic(S25);
IsQuadratic(S625888888);
: x/ W) Q% r+ ?8 l) O( vFactorization(w^2+50);  
Discriminant(Q5) ;
2 d( ^* L4 E+ w" D3 sFundamentalUnit(Q5) ;
FundamentalUnit(M);
" b4 H3 R) s5 x9 k; ^Conductor(Q5) ;
5 U6 \3 h( U( a
2 }5 J8 K# [. Y( r  m2 ]Name(M, -50);
1 o, ]/ K# [0 tConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
0 Y+ b/ t" {8 G4 `- ^ClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
5 \; c, V% Q9 D/ W  K# zPicardNumber(M) ;
! I% Y8 f: m; s( O
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -50) ;
0 A- n, X7 q3 gNormEquation(M, -50) ;
; P5 T, s# u# d. b2 f' P
! F: H: t7 ]1 EQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
9 n# t" c- X& v1 q2 ?Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
! ~0 X6 k& p4 c8 f+ K$ sOrder of conductor 625888888 in Q5
4 d: h, W0 t# O* ntrue Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
true Maximal Equation Order of Q5
' `2 b4 k  f; t4 |9 |true Order of conductor 1 in Q5
+ W2 N8 `  B) ?true Order of conductor 1 in Q5
% Y5 K1 ?' r, J! V% ]$ v* _, H: R% L, htrue Order of conductor 1 in Q5
[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
-8

- S4 z5 c' Q1 b) }& v4 y>> FundamentalUnit(Q5) ;
0 j" ^8 Y& |5 q. n2 G                  ^
: _+ Q: j+ L# L7 n) W; WRuntime error in 'FundamentalUnit': Field must have positive discriminant


! c! \% U4 _6 N: d, M/ C8 m>> FundamentalUnit(M);
4 v9 O1 j: k" {' J' F# X: {' V                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
/ |$ w: c1 R. B
8
* m: m8 p% {3 U
( F* M2 n1 S' K" J>> Name(M, -50);
       ^
1 j0 U7 B4 V8 ~+ ~Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]

, i2 Y- _" Q) C6 Z" ?8 [+ F1
6 u( g& K7 l# R5 K1 F" FAbelian Group of order 1
3 I- m6 X' m5 [0 J2 M0 }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
( b5 t2 z# C) K- ~7 r' M- o7 N1
' M9 l6 z4 o, {1
. k# r6 O) i' }# ?4 t! r" L5 xAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
1 u, K8 R. ?! ~, l& p$ O-8 given by a rule [no inverse]
Abelian Group of order 1
7 H. f" f8 I  \: d& AMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
false
5 ^) S  |. W: H5 Sfalse

lilianjie

看看-1.-3的两种：

Q5:=QuadraticField(-1) ;
8 C& H; `. u+ N2 |! W- Z: p  fQ5;
) N8 [* A6 M& L1 y! U- j
+ ]# \: K  i; y+ R# bQ<w> :=PolynomialRing(Q5);Q;
# A. {! F; E3 L# ZEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
2 S* q; C0 W3 E) T! W2 G) S9 \M;
- ^: j  a& t8 f$ gNumberField(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;
, C* Z+ m4 Q! R: iIsQuadratic(Q5);
( D7 \; d' s* r$ {' r$ a# ^IsQuadratic(S1);
IsQuadratic(S4);
9 V  d; d7 p' m9 G: w  ~% v3 kIsQuadratic(S25);
  }" t$ B+ ~/ p2 s( z/ I. c8 oIsQuadratic(S625888888);
" ?, I, F# \* q! }0 vFactorization(w^2+1);  
/ ?$ L' ^0 _# v& J5 x0 XDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
1 X3 Y' m2 l! n: u6 ]Conductor(Q5) ;
' @1 ]* Q% b/ ~! M* Z7 D
. Y' Q7 C' E, R0 d6 yName(M, -1);
5 ^. v1 {3 L7 K, D: |- sConductor(M);
) {, o; P% y% f2 s- M! c: JClassGroup(Q5) ;
ClassGroup(M);
: U5 J$ u) p5 C' F( b1 OClassNumber(Q5) ;
ClassNumber(M) ;
( }6 a2 k/ {: Q# X/ zPicardGroup(M) ;
PicardNumber(M) ;
8 a8 h2 }7 \/ U" v3 A8 k
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
' Q! [7 {! J# d/ i  F' ANormEquation(Q5, -1) ;
8 M5 n$ G8 C9 t  y4 N  L% }NormEquation(M, -1) ;

! {0 }: f! I$ G: P; m4 oQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
- J6 D$ h* a% [: E* jUnivariate Polynomial Ring in w over Q5
1 I4 w  u4 H6 r6 D4 M9 W0 U: Q! hEquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
4 [8 i' G  c( e" D, p5 _" _) EQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
2 g7 N& v' ?! v3 y9 @: c0 }Order of conductor 625888888 in Q5
1 c0 Z7 @! S4 M& p5 ytrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
* u2 C( q  L' A" x  S* H* r; s7 Ptrue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
. N. O8 }' w* D" r- P; u3 L[
- N7 K$ k& [9 o4 Y& k" {    <w - Q5.1, 1>,
0 _5 R% c! \  e    <w + Q5.1, 1>
% W) L+ T% E$ m) @# o3 U- a1 U]
-4

>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
+ m  s+ @$ Q2 U  O

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

4

>> Name(M, -1);
: e$ Y: P6 {* _       ^
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]
9 q1 b6 j$ i" N* g* o
( v( d$ n& z6 V( {9 ?( L' W1
. t: U# P( j  H4 i8 R5 KAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
% d" z3 W* _( t7 d! w/ ^. UAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
* C( o, Z2 U& W  p1
8 Z5 T2 P# A  F, e- p5 y4 {  [. VAbelian Group of order 1
" M% G5 |; J$ \; [" \# v2 jMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
( q' t# o6 e6 A2 Minverse]
' C6 x% r9 Q5 U3 e; m5 {1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
  F' y. t$ g. W; B4 |2 E-4 given by a rule [no inverse]
Abelian Group of order 1
* p' j* a. ~+ \Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 given by a rule [no inverse]
false
false
===============

. M' ], \. h  Q( Q/ fQ5:=QuadraticField(-3) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
NumberField(M);
4 a- v: n1 J  [' RS1:=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);
7 ?: Q$ J/ b: y( h! w! C; lIsQuadratic(S4);
IsQuadratic(S25);
% G4 X3 T1 ~- N0 ?IsQuadratic(S625888888);
Factorization(w^2+3);  
Discriminant(Q5) ;
, w& y+ c* @' S% p  GFundamentalUnit(Q5) ;
FundamentalUnit(M);
' B) Q: g* a" w, q2 e, N7 tConductor(Q5) ;
$ \" r, E7 c  G
0 c" E* Z% D  _. ~( t7 wName(M, -3);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

. T6 U) p$ @5 ?' H3 E$ dQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
, p& L; O0 c5 t4 I: k) ?% eNormEquation(Q5, -3) ;
NormEquation(M, -3) ;
: D! f& e" j+ `5 q; ~1 ^
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Univariate Polynomial Ring in w over Q5
" c( B$ v' ?* Q) m; z+ E. b5 N. hEquation Order of conductor 2 in Q5
+ }' ?7 a* y5 {2 g$ V# I: `+ B0 t' iMaximal Order of Q5
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
- I1 N3 h9 X' }- a0 I; Ntrue Maximal Order of Q5
true Order of conductor 16 in Q5
true Order of conductor 625 in Q5
+ C' k7 R& l4 K7 x6 atrue Order of conductor 391736900121876544 in Q5
[
6 a  a4 @$ z- H* ]" a2 X    <w - Q5.1, 1>,
    <w + Q5.1, 1>
) a( o$ N' g  _8 r]
-3

3 @, S1 [2 v6 T. v>> FundamentalUnit(Q5) ;
                  ^
# e8 j4 I! l( U+ {  CRuntime error in 'FundamentalUnit': Field must have positive discriminant
% B0 W$ i5 [, O; a
% h7 A  U: E5 U9 e+ H
>> FundamentalUnit(M);
                  ^
0 o! o! d7 ^) U, a' P; MRuntime error in 'FundamentalUnit': Field must have positive discriminant
" ~9 X8 i) O  h3 G3 }$ i  t3 C
% ^( \& S+ s, z& C3
' K8 C1 J. _# g9 F& s3 }
>> Name(M, -3);
       ^
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
4 I" T9 d# s  O# x4 R
1
% O+ g  v' W/ k, B) L# rAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
/ i& d* Q4 n+ b4 E. }6 qAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
5 y( Y/ P5 O1 f! m* z1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
4 |; v  y/ c1 B3 ^, H1
Abelian Group of order 1
7 A" B6 v# W" tMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
0 Y  J0 _3 h4 d' A" w% k) _-3 given by a rule [no inverse]
: Q0 i  }- A$ mAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule [no inverse]
false
" G9 x% O( m0 t3 c8 ]. Q: c+ Sfalse

孤寂冷逍遥

lilianjie

# H4 w! U; D- W, U+ S8 x5 p( VDirichlet character
Dirichlet class number formula
+ O8 K0 L2 E% B
% i) O! L9 J0 Z3 [7 ?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

-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

-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
h=-6/（2*3）*Σ[1*1+（2*（-1）]=1
) A+ e" l: }7 y; M2 l; a3 B( ]( a- u: I
1 p3 g6 F8 F7 _7 b-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，

/ N  k+ x  }' a3 f

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



-50时  个单位根                          N=200

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

  m; a8 h" {% r. iF := QuadraticField(NextPrime(5));

  C- Z( U8 f1 {3 oKK := QuadraticField(7);KK;
- z5 o& k$ y" s- VK:=MaximalOrder(KK);
Conductor(KK);
7 j/ Q& a& K+ U5 T1 Q7 R( f: }ClassGroup(KK) ;
" m- w7 k9 D) l& D$ TQuadraticClassGroupTwoPart(KK) ;
NormEquation(F, 7);
A:=K!7;A;
B:=K!14;B;
Discriminant(KK)

* J# J: V% \% D3 q! ?% ^% z' WQuadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
! a6 v. g& ^" u$ B) b- Y28
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
! V6 d3 x3 a9 l# YRelations:
3 w% Y5 o  r: h# P, X* \    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
+ v. P: d+ T7 W  Z0 D7 uRelations:
& L/ F2 a6 _: z+ ^( K; X    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
  k: L# Y" Y0 y4 yinverse]
false
7
8 F0 ]7 l. ^( k, m! R14
28

lilianjie

7 {( x1 i) @& T6 [






分圆域：
C:=CyclotomicField(5);C;
& |+ i7 N- u" H5 SCyclotomicPolynomial(5);
$ ^) P0 I* P* d* ^9 KC:=CyclotomicField(6);C;
: V3 ~1 Y" O0 |9 E# {) WCyclotomicPolynomial(6);
5 S- j7 {' U( B- y( d! [. vCC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
) v' K8 `7 L) L4 a2 hMinimalField(CC!7) ;
MinimalField(CC!8) ;
MinimalField(CC!9) ;
! {. T. v; W# R. z( _" F. s) oMinimalCyclotomicField(CC!7) ;
: j7 A2 E! t7 o! }- u' u, t2 ERootOfUnity(11);RootOfUnity(111);
Minimise(CC!123);
& u* J7 G. Q" E) V! ]& NConductor(CC) ;
CyclotomicOrder(CC) ;

CyclotomicAutomorphismGroup(CC) ;

Cyclotomic Field of order 5 and degree 4
7 [- P7 S7 N3 C$ k; C! }$.1^4 + $.1^3 + $.1^2 + $.1 + 1
, a( D% Y% N; c% t! tCyclotomic Field of order 6 and degree 2
. A  ^1 Z& ~( B$.1^2 - $.1 + 1
Cyclotomic Field of order 7 and degree 6
3 f  R- l* P' A" L% X6 l$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
Rational Field
; }  ?" n/ C+ eRational Field
& `& h' t1 N0 ?Rational Field
1 Y+ G2 {: z; A+ w# K" [, Yzeta_11
zeta_111
  m* x2 Q- `' e( Z8 {3 x6 Y123
: h6 K) j; G# R7
7
4 _$ d8 O3 @/ B$ `Permutation group acting on a set of cardinality 6
Order = 6 = 2 * 3
" |) t4 v0 R& F$ v2 N. L6 L    (1, 2)(3, 5)(4, 6)
/ V" q' Y2 s2 K) e+ p3 E$ [- c: j+ {    (1, 3, 6, 2, 5, 4)
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC
Composition 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
6 P; a! ~  ~1 tCC

lilianjie

lilianjie 发表于 2012-1-9 20:44
% J  H% D1 w8 z& t8 L) p- n分圆域：
C:=CyclotomicField(5);C;
4 v0 X; f0 B" Q$ ^: n& SCyclotomicPolynomial(5);

7 I( N9 C1 K& D$ K2 z分圆域：
( c2 C* Q  b8 `分圆域：123
( W; g/ R3 u; k0 R5 b. ~
4 F) b1 y; L# YR.<x> = Q[]
/ @% ^) C, P6 x/ LF8 = factor(x^8 - 1)
F8
1 h" d& p# W0 n# Z
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

7 V; U2 f3 h2 u& wQ<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
4 ~) m( Q& i) _' A. v* xFF:=CyclotomicPolynomial(8);FF;

0 B  {& }+ K. F" ?' L4 eF := QuadraticField(8);
F;
6 L4 E6 A4 y2 UD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
2 r' V- |) k% |6 U8 ICyclotomic Field of order 8 and degree 4
$.1^4 + 1
% K/ R. N* `' UQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
- {4 e6 y* s5 N/ O; P6 _[
    <$.1^4 + 1, 1>
]
  Q6 j4 r% X1 c/ J! c) ?
R.<x> = QQ[]
  ]: Z  }6 N2 C& \  ~F6 = factor(x^6 - 1)
2 C9 P9 e9 t# S) \# {) qF6

(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;
% {2 w0 f, d" G8 G: E6 rC:=CyclotomicField(6);C;
( Q9 {$ G  q& e2 Z( zFF:=CyclotomicPolynomial(6);FF;
2 d9 a& T0 f, C( W5 y8 W
9 o' {0 o4 Q3 ^F := QuadraticField(6);
' ]0 ^3 a' v7 j* IF;
  x4 e+ L' T9 f9 M' v. e, ?% PD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
7 G+ z' A4 |- N( S( K" y- HCyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
% G. D7 l% H+ A: g, `* ^3 OQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
    <$.1^2 - $.1 + 1, 1>
9 s3 Q4 M5 t8 N/ m) U; J( e4 b: r, r% ]]

$ t8 Q  z; Z8 D9 D# J/ BR.<x> = QQ[]
F5 = factor(x^10 - 1)
& I& F* m2 J3 H6 M! m) BF5
0 r/ s' P! S  B2 S1 `7 o2 k(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
! g$ c) T& m. a% r, p2 p
Q<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
, v2 @4 ]! m8 h  K3 v& ~FF:=CyclotomicPolynomial(10);FF;

' ]; P6 E7 Y, H- \F := QuadraticField(10);
F;
D:=Factorization(FF) ;D;
8 U% X0 G$ f" sQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
- {% M4 b6 h0 x% ?" U8 P1 ?2 u8 ZCyclotomic Field of order 10 and degree 4
8 N: s/ S6 e; o1 Z1 v$.1^4 - $.1^3 + $.1^2 - $.1 + 1
2 t. ^8 i5 t5 o" G+ TQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
- R( F4 \' l4 L2 D# i[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
( l8 m6 A# w. c# c' [) S8 C. A]

lilianjie1

分圆域：123