虚二次域例两（-5/50）

lilianjie

) Z/ J7 x) J- X% e3 N0 \Q5:=QuadraticField(-5) ;
0 L+ x" D! w* {/ j7 oQ5;

7 D* T) N& l! R7 }  [, uQ<w> :=PolynomialRing(Q5);Q;
' f3 c! [+ [2 m1 f, b# Y; OEquationOrder(Q5);
& t$ s7 q4 B, n% C+ xM:=MaximalOrder(Q5) ;
8 u9 r4 G% D2 [, _1 dM;
5 Z7 s6 p/ d$ x% o% g0 vNumberField(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;
$ b5 P$ o! N. T9 p2 s8 o3 UIsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
+ D4 S4 Y# a# fIsQuadratic(S625888888);
Factorization(w^2+5);  
: L0 f7 j4 S0 ?% \' \. _  UDiscriminant(Q5) ;
( s+ @( N) C" y3 DFundamentalUnit(Q5) ;
) Y0 {/ s& @. B4 w' |! o; L. UFundamentalUnit(M);
+ O5 x9 B8 D% i  r3 F7 BConductor(Q5) ;
6 Y9 R+ z; e! U, H# H0 n
Name(M, -5);
, l: B& t% U; p* q2 b$ `0 }" V  }Conductor(M);
ClassGroup(Q5) ;
2 a  q) N: i1 m! [2 X) k8 R$ G5 wClassGroup(M);
ClassNumber(Q5) ;
1 j; @# C/ K/ f3 L% J4 zClassNumber(M) ;
- v4 ?* F2 e; _! ^4 d- `PicardGroup(M) ;
PicardNumber(M) ;
& A: p6 A$ K2 S  n4 b) J
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -5) ;
NormEquation(M, -5) ;
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
8 d! l& h5 H1 o8 m% z$ m' j: U- q' bUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
9 x) P8 w6 |8 ~) QMaximal Equation Order of Q5
3 t+ P$ h' l' N. R. x/ gQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
, k3 v# v4 V# e& y+ k# Z0 NOrder 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
! @+ c* {8 s+ O( N0 _true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
9 O6 v, B! y" q    <w - Q5.1, 1>,
' N* G1 k  o1 z    <w + Q5.1, 1>
]
8 s  i5 N3 Q* C; q. z9 P8 k-20
5 H- i2 H" O/ k
>> FundamentalUnit(Q5) ;
; {% i* m# @, h# r) Y                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
5 n3 O8 [  L0 l% Z
/ e3 a5 X, d) \: D
. |" t5 Z1 z, }* l  t3 [>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
8 b6 k  j0 a6 d" I" r' a
20

; a* h; Y- O# b4 e7 \>> Name(M, -5);
* o) m! b/ Z2 O       ^
Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]

1
/ t/ Z9 y& Q# cAbelian Group isomorphic to Z/2
4 X* K0 }  o. R7 Z& S7 iDefined on 1 generator
Relations:
) J7 P+ Q8 C" R6 _& [    2*$.1 = 0
' J7 F, Y8 |* t5 QMapping from: Abelian Group isomorphic to Z/2
% f) }/ J) {# l( DDefined on 1 generator
6 X  z: o, a5 v" z, i8 Z/ PRelations:
; R- c3 V; I. f! ^$ t$ r    2*$.1 = 0 to Set of ideals of M
; s- z6 K9 Q4 G8 k1 iAbelian Group isomorphic to Z/2
$ b6 Z# M: ?4 F# b0 r. pDefined on 1 generator
* Z& [  K& G' [( C+ P& lRelations:
1 j% S  g/ O8 M! G# o    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
$ W6 E- `6 `! T( o# BRelations:
    2*$.1 = 0 to Set of ideals of M
2
4 {  U& S- G& S' s0 ^/ K! n0 w2
! M' _5 n7 s! l! F  e+ \Abelian Group isomorphic to Z/2
Defined on 1 generator
, p) v$ L* t, X5 ^7 ]  T/ gRelations:
$ Y0 u6 ], q' d- c, Z# G5 y. t" v    2*$.1 = 0
& k! C( ?4 o$ k$ KMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
: S) y! q$ N8 q0 Z; R8 URelations:
3 f3 u" @$ Y, a9 z7 U    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
2
Abelian Group isomorphic to Z/2
Defined on 1 generator
5 p. G( J* C* p7 S% V; s0 `. eRelations:
' }3 L5 t3 M2 h% L0 g6 `    2*$.1 = 0
0 ~3 ~+ f$ `$ L) G* i) K; RMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
% x7 U/ @- n0 C8 i, R$ N    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
0 I, I$ @" s4 V3 M% Y* tAbelian Group isomorphic to Z/2
: f3 Q7 J# ]; |6 {2 SDefined on 1 generator
( I3 A7 @& w9 D! l4 cRelations:
& a7 ]5 k1 p0 S7 W1 F    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
& ?/ w* z! k% B6 b1 A. P/ cRelations:
- a% C. H* o+ `' C5 U    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
false
# F& l. X) D# [0 cfalse
==============


Q5:=QuadraticField(-50) ;
6 ~1 |* ]! X( O3 WQ5;

9 v  Y$ K  N$ u  P; A! gQ<w> :=PolynomialRing(Q5);Q;
6 A6 W! Y" v5 i' t! E, L6 uEquationOrder(Q5);
6 p" p% U  }/ p# x' n! J; SM:=MaximalOrder(Q5) ;
M;
% F7 _- {0 I* n( @NumberField(M);
0 Y" q1 t4 H* A% WS1:=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);
! _/ @& D/ v+ Q! P. _IsQuadratic(S4);
& H" }# \; Z# C2 LIsQuadratic(S25);
8 @7 u/ m6 W5 ?8 ^$ ?IsQuadratic(S625888888);
Factorization(w^2+50);  
+ a* X- }$ O, Q' }! i0 y- fDiscriminant(Q5) ;
+ J. ?3 Y1 ]2 t/ a$ zFundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
0 E& @9 e1 b) d( `/ h
Name(M, -50);
) h6 R* q% l5 A! z1 nConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
* z7 r7 i3 y0 f2 m+ [, NClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
5 M7 N6 D% z& Q. K: |/ cPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
6 F& `* ~& n/ C% eQuadraticClassGroupTwoPart(M);
- L2 ]9 I- A+ W8 I& p+ _$ ENormEquation(Q5, -50) ;
5 k6 q$ ?" z# q# V' _& C9 oNormEquation(M, -50) ;

Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
! S5 }6 f- O( c. D1 i- yEquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
' w" ?2 X9 B% n3 G" d* BQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
- I( z$ I% [4 |& ]) z) JOrder of conductor 625888888 in Q5
7 Y* ]9 c' W& y; I) B. \true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
' g$ G7 D* x+ f$ p# H/ \2 ctrue Maximal Equation Order of Q5
" }, l) R( y8 z7 P+ i/ f! x0 ~  rtrue Order of conductor 1 in Q5
. `3 g1 e8 [. J- atrue Order of conductor 1 in Q5
" ~; i! `8 {% p5 H2 Z2 htrue Order of conductor 1 in Q5
[
  D/ ?' M! ?' L. ]# {    <w - 5*Q5.1, 1>,
- M/ h) b$ p- \: R: G2 i: u    <w + 5*Q5.1, 1>
]
; \: I9 T+ X& o6 d& Y: T-8

>> FundamentalUnit(Q5) ;
- p5 n5 |/ K" i7 P                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
8 T0 g" S3 p3 n
3 Q/ o& n  a8 H) Y2 Z" c1 B
3 h0 g" M$ ^& P: f>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
5 P6 S& n% u2 T; \2 D( t
. c/ J" ^1 X( @  I8

! U% u7 L1 e. P( M>> Name(M, -50);
       ^
  x* M- _0 V* l5 X- RRuntime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]
, `1 V! y! g6 H2 m. r
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
8 S0 r2 q' o% T1 \Mapping from: Abelian Group of order 1 to Set of ideals of M
1
. d3 X! R9 C4 v) r5 s* }: }; P1
, m# h  e4 O9 C$ }* G+ b' j: MAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
) q/ c7 e4 H5 }5 l. L" winverse]
1
( z3 Y: G" O- H& F5 P" L: n. S7 T; AAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
) |6 s9 O) N4 n  x/ V2 pAbelian Group of order 1
; V. p0 s* d( v$ x5 x8 j1 x  K8 ^Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
0 B5 V, _% `7 @false
false
  m! z+ B- l7 T

lilianjie

看看-1.-3的两种：
. \% Y; M3 a* r* d( f$ f# i. T
7 T) ^6 S) r1 k+ ^" XQ5:=QuadraticField(-1) ;
1 U5 Y8 h. A- n8 |. ]2 {; PQ5;

6 F9 r. L5 O* \6 \  EQ<w> :=PolynomialRing(Q5);Q;
: H" Z, u3 z' PEquationOrder(Q5);
9 X6 p5 n. \8 Z5 \" U( G* F$ E8 DM:=MaximalOrder(Q5) ;
M;
NumberField(M);
1 z8 m7 l% d% ~7 TS1:=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);
; `8 ?# t5 j; h: ?Factorization(w^2+1);  
2 `0 i. z2 Y9 s! y, K. vDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
) e2 {" W8 j) \/ J, w/ Q2 t, h# rFundamentalUnit(M);
Conductor(Q5) ;
: G8 X8 x" N3 _9 D1 G  L1 U- [
Name(M, -1);
Conductor(M);
ClassGroup(Q5) ;
9 a" t+ {+ ?$ s' I( s# \ClassGroup(M);
- O+ e5 l4 l2 @ClassNumber(Q5) ;
$ K- a3 r0 J/ m3 I& AClassNumber(M) ;
& o5 s9 s; V8 p7 Z7 i, H5 }PicardGroup(M) ;
; c! G3 o) c" nPicardNumber(M) ;
/ S, K$ P8 a( O& {% m
2 m- ^% u* P& M; W( FQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
  i( |! L: B- t' SNormEquation(Q5, -1) ;
/ W7 V5 M/ X: q8 I  a( iNormEquation(M, -1) ;
1 _% c# C2 I2 i: U! v0 s8 p
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
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
, f( Q- Q3 h: R7 _7 `7 u5 n5 W2 nOrder of conductor 625888888 in Q5
$ K8 `: I3 i+ h2 `/ z. l9 Ytrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
6 c& r- ~" z2 r  Q+ l/ P8 Vtrue Maximal Equation Order of Q5
% Q2 M: c8 k4 S" x! n/ F4 X! dtrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
7 @4 n7 J0 B6 W* m3 V5 _- r[
% H, a  J9 e, H0 I' S    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
-4
* ?7 i2 n# N, f& n# e' ^* ]+ r+ r
>> FundamentalUnit(Q5) ;
" X: E* o( j/ [9 o' N  T                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
3 b/ H1 d6 S, L4 [$ T- y/ h. [

>> FundamentalUnit(M);
* v4 m' A6 _5 H, F6 r                  ^
4 ]! d+ O! F0 P2 S5 s/ j0 }) t4 W7 iRuntime error in 'FundamentalUnit': Field must have positive discriminant

6 M' u6 _1 E" B4
( n; u& k6 V5 B6 o
>> Name(M, -1);
5 a& f$ v9 H7 ?; f$ H       ^
& M7 n3 ]& c# b3 _$ URuntime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]

1
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
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
- t( ]) f! V1 R7 {inverse]
- y2 F, {! y5 k1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 @; \% f. a+ H% ~; Z  O/ c-4 given by a rule [no inverse]
  @1 O: l& ^' l$ n  k8 e& D  XAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 a( L" h$ ?7 V4 }-4 given by a rule [no inverse]
false
/ P# R) s- D# `3 vfalse
===============
& {4 J, r7 ^4 _( Y, E4 }
Q5:=QuadraticField(-3) ;
2 c8 ]$ W  n1 zQ5;

Q<w> :=PolynomialRing(Q5);Q;
9 s0 ?; p( V5 d; v3 @: K- vEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
NumberField(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;
7 q3 c- s7 Z" t% q4 GIsQuadratic(Q5);
6 Y& E4 \5 P3 t6 a& q" u- ?* WIsQuadratic(S1);
, Q7 w  K$ t" VIsQuadratic(S4);
0 R, r/ x8 h/ P. }+ B& {" O# L' dIsQuadratic(S25);
# p: y" _( S2 v  z4 j% {( {( MIsQuadratic(S625888888);
Factorization(w^2+3);  
! W, ~- U9 F$ I" \# kDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
  E9 S5 F# D& V/ N
" z4 q: I' I" h7 CName(M, -3);
Conductor(M);
ClassGroup(Q5) ;
/ F& a/ G( }1 c* ^% X' j2 bClassGroup(M);
ClassNumber(Q5) ;
, p) [. K9 T* z. P2 Y$ xClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
2 Y# k0 c, i2 X. a# b
QuadraticClassGroupTwoPart(Q5);
* U; ~: a9 D& @" ZQuadraticClassGroupTwoPart(M);
/ z0 z) B( P" ^5 w+ l' e' GNormEquation(Q5, -3) ;
+ {2 |- y3 _' S/ rNormEquation(M, -3) ;

Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
5 }4 E4 @% h) I# n% c) B8 n" B4 A! t9 kUnivariate Polynomial Ring in w over Q5
& t# ]5 M+ u6 G( L+ M6 t! |: zEquation Order of conductor 2 in Q5
- [5 B' @3 m3 z: uMaximal Order of Q5
( Z/ w' j, T9 C/ jQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
5 M6 ?) [* `5 [6 v9 g0 U) VOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
  C7 q  [) ]) Z) o5 Z5 I! vtrue Maximal Order of Q5
8 i; Z: f8 X+ ]% v. Dtrue Order of conductor 16 in Q5
true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
9 u8 p+ c# V; ?5 L-3
& }2 \! C7 I( O. w# E8 k; X. M% G
! k0 ?6 t4 }: r+ p) j>> FundamentalUnit(Q5) ;
$ Z8 ~7 f+ [; H' ^$ {                  ^
8 S# V9 _, e8 a" Q) [/ d" ]Runtime error in 'FundamentalUnit': Field must have positive discriminant

* u; N2 M4 Y" f. z- ?7 ]( I+ L4 }6 n; ^
>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

3

>> Name(M, -3);
       ^
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
: w8 M# q3 a+ E0 u6 N' c
1
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
9 o5 ~2 n# y, \1
# E7 S0 q4 V* d1 s: _1 o( |1
8 m$ v) e) [+ E0 I, z' DAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
, G) l, c+ d2 Z4 Q7 n3 q2 m% ginverse]
1
Abelian Group of order 1
/ K: W9 a, s# p, t( A% l. q. j% JMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
. R1 |1 E1 a% u" W& {! \8 a-3 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule [no inverse]
/ B$ }# h' o1 O8 L% K! Vfalse
false

孤寂冷逍遥

lilianjie

! }- t! P8 \' b
Dirichlet character
Dirichlet class number formula

; ^0 N( U+ p$ X6 o( O' d, M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

: U9 c; T0 ~$ V' T3 X-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

-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，
4 n% c" f2 S2 P5 o9 {
& R; e+ s2 [8 ]; E7 ^- Z! e

" L' K5 U. F% ?& Kh=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2

: x- M' ?2 `- c' F3 `+ X3 z7 C
# I( T3 b; ]* l7 ~- k- B+ Y
-50时  个单位根                          N=200
3 Z4 p6 M" p# g. a: A) U

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

# _* [1 h. v2 F/ }. P3 EF := QuadraticField(NextPrime(5));
0 `; z  {0 b+ }5 E
* _/ E1 f: `7 F* F$ ^& VKK := QuadraticField(7);KK;
6 M9 X) a5 l/ G$ _* C/ T. `K:=MaximalOrder(KK);
Conductor(KK);
! C0 [; q  X1 Q7 N( g- q) t  iClassGroup(KK) ;
QuadraticClassGroupTwoPart(KK) ;
NormEquation(F, 7);
A:=K!7;A;
9 v8 c5 r4 y4 U7 I+ kB:=K!14;B;
) ^3 q3 c2 o& E3 }0 t5 SDiscriminant(KK)
1 ~$ b+ r: K- q6 v  T' [
Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
" I3 |. E7 p- i" v28
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of K
" l& q( W( J. T+ C! B/ CAbelian Group isomorphic to Z/2
6 W$ W, m7 t, N; pDefined on 1 generator
Relations:
6 [7 \( J1 F# N. }/ I8 J    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
: C( r) U& h# {' v( G. jRelations:
    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
2 K% i* Z) ~8 T4 U5 |; Binverse]
5 p) x1 k  T) I6 J" _false
- l5 w2 \. O' v+ P9 d7
& x2 a/ R1 U, E) {14
28

lilianjie

0 q8 j2 S- n/ d& X7 T; w: D+ \
( a, r* [$ Z8 t' o$ q7 e0 W! \
- @/ \5 ?, s+ \* _- d  W4 N
) L( Z' z4 G% o1 W

4 |5 z2 ]% h" E, ?3 s8 ^" ?" ~
0 S) p$ A6 }& {分圆域：
- n1 x6 |5 K9 ^0 Y$ XC:=CyclotomicField(5);C;
CyclotomicPolynomial(5);
- K! f7 `( s$ `C:=CyclotomicField(6);C;
- k3 [% y1 H, cCyclotomicPolynomial(6);
7 Q2 i7 Z& m7 D( aCC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
MinimalField(CC!7) ;
; H, o7 y" G) K+ d+ ]8 E6 q9 UMinimalField(CC!8) ;
MinimalField(CC!9) ;
MinimalCyclotomicField(CC!7) ;
RootOfUnity(11);RootOfUnity(111);
9 A# `( i! k( ]: L, U9 B1 GMinimise(CC!123);
* t* w6 W7 ]  U7 m" p* I3 YConductor(CC) ;
CyclotomicOrder(CC) ;
& p4 s2 I9 ^1 r( m( f) h
4 i0 D/ P5 K6 V5 o; A5 C. y) [CyclotomicAutomorphismGroup(CC) ;

Cyclotomic Field of order 5 and degree 4
+ i7 s( Y/ ~$ W$.1^4 + $.1^3 + $.1^2 + $.1 + 1
Cyclotomic Field of order 6 and degree 2
. a9 L0 Q  g3 j7 P: d  {# y$.1^2 - $.1 + 1
Cyclotomic Field of order 7 and degree 6
- |% `; w3 l# M  i! l- L3 J$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
0 C& z* m7 b" _% o2 ^. C. zRational Field
6 g* |0 k% ^9 |' }1 Z9 f' mRational Field
Rational Field
3 c( h+ C. ?0 m/ Rzeta_11
zeta_111
# E8 z) D( @/ n) U' @$ N$ j123
5 v7 J7 A5 ^% K5 I* l+ d7
  @5 z  Z+ a8 F& C: X8 i' `" [7
  C7 ]- A- H$ c+ YPermutation group acting on a set of cardinality 6
Order = 6 = 2 * 3
    (1, 2)(3, 5)(4, 6)
. S0 b$ H' {+ _1 ]0 U    (1, 3, 6, 2, 5, 4)
+ ?9 g2 H( K9 y$ N/ I9 P# M% `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: $,
6 o, ?8 e9 _5 I+ k! LDegree 6, Order 2 * 3 and
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC

lilianjie

lilianjie 发表于 2012-1-9 20:44
分圆域：
C:=CyclotomicField(5);C;
- A. k: k, T) Q# uCyclotomicPolynomial(5);

7 x2 f! C5 G! N" k# J, ~
分圆域：
分圆域：123

4 g0 F0 @8 ]( m9 qR.<x> = Q[]
- g! }( ?: P4 s/ m( @F8 = factor(x^8 - 1)
F8

3 H7 L& l$ [+ [5 n/ A3 H(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

# ^7 ~$ y, N' e1 V7 \Q<x> := QuadraticField(8);Q;
$ m$ ]5 p& h/ O  G2 j9 YC:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;
. p6 T6 C" w" m* U+ i0 W
F := QuadraticField(8);
F;
. c9 ~" N# v. kD:=Factorization(FF) ;D;
: `5 ?# Y+ X1 T4 R& Q2 M$ t1 QQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
1 `( S' v# k. m' YCyclotomic Field of order 8 and degree 4
$.1^4 + 1
$ c; @- Y1 Q2 Y8 |Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
/ y- H3 K) k% }& _    <$.1^4 + 1, 1>
4 i# [4 W2 p* Q0 p9 |9 {& G5 s7 B]

R.<x> = QQ[]
F6 = factor(x^6 - 1)
9 Z2 T" j( s$ N- E) x( yF6
5 l* G* Y+ u/ ?! o5 w
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

3 ]/ i: ~5 ]+ n6 c! yQ<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
9 r: ~3 d- z3 G- ?  vFF:=CyclotomicPolynomial(6);FF;

& M& a4 I$ ^/ UF := QuadraticField(6);
+ n6 x  p8 G4 g4 {& X  aF;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
5 h2 f( a3 e) q7 J+ o2 h9 ?) e5 ?Cyclotomic Field of order 6 and degree 2
- a0 F: _' @% d$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
    <$.1^2 - $.1 + 1, 1>
% h. f, ]/ [, l2 s) v# \+ H$ Z]
: y' ]8 L2 t* U
3 Z. r6 l" {& ~5 ]6 gR.<x> = QQ[]
4 f* b9 n3 O: [) i! u9 Q: uF5 = factor(x^10 - 1)
$ z" [; _2 i# J8 x* c: RF5
/ L" ]& w+ X) }: U% x(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
/ }- b- U; H8 B7 J4 ]1 K" r1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

Q<x> := QuadraticField(10);Q;
6 L% G' ^) c% SC:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;
: V7 z6 ]4 @  ?! Y( p: C2 J) B
F := QuadraticField(10);
9 \5 D6 M9 z$ i2 F0 ]- y" wF;
: c: l) c4 ?4 I, g: T1 a& kD:=Factorization(FF) ;D;
2 ?6 V. D9 z" o: v, U& b' SQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
9 D. r, D! u6 U[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

lilianjie1

分圆域：123