虚二次域例两（-5/50）

lilianjie

Q5:=QuadraticField(-5) ;
3 o: {, M6 d4 H! j  E- j! VQ5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
' C. ]- L) p* ]; s" SM;
5 Q5 V( h* o, P! y8 n. kNumberField(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;
, z" ]( d' A. ^IsQuadratic(Q5);
8 P8 l( X- B8 Y8 H2 U6 xIsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
" @0 q/ [/ p" P, B( `IsQuadratic(S625888888);
& W; f. I6 ]! K8 y( K8 ^' rFactorization(w^2+5);  
Discriminant(Q5) ;
- X& R. e4 \5 Y5 j/ l: z) XFundamentalUnit(Q5) ;
FundamentalUnit(M);
- s- t% x( C! @( U$ R+ V. yConductor(Q5) ;
) E6 I- N% e3 a; {
Name(M, -5);
Conductor(M);
4 D6 G$ R* B1 A# A2 X( `ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
1 U/ z6 e. k& WClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

4 C3 ^+ L. O! d( _/ i4 K/ i1 ]QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -5) ;
: t  I, y2 U2 n8 N9 r- \: @; g) ANormEquation(M, -5) ;
# Q+ v3 c% Z" LQuadratic Field with defining polynomial $.1^2 + 5 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 + 5 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
' e  m9 w# T. `. H: {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
/ ?' C3 r, _) n/ h8 Z- m0 g# V[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
6 e+ E% C: {0 k]
-20

>> FundamentalUnit(Q5) ;
                  ^
, E( t. ~2 a$ E, U8 ~9 ]Runtime error in 'FundamentalUnit': Field must have positive discriminant
9 F- g$ g" |/ Q5 T: [

6 P$ B% V+ z, a' w/ _: A>> FundamentalUnit(M);
                  ^
' ]/ E1 {5 h' @Runtime error in 'FundamentalUnit': Field must have positive discriminant
2 ]7 k. V8 P% B1 [3 \  W
20
( T  r% S- [. c* Y* _9 T
/ p! V4 e9 J( U) \6 ~# v) L- h8 B>> Name(M, -5);
       ^
Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]
% W8 A4 w% z1 p% {3 m6 E- U
1
Abelian Group isomorphic to Z/2
4 ^/ ]4 v% @: V7 u0 v# sDefined on 1 generator
0 `2 c0 W0 [, @Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
, P: o1 T4 I8 M5 j( @  h% @Defined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M
Abelian Group isomorphic to Z/2
( @' ]0 G5 o) L4 p( k8 y! f* WDefined on 1 generator
Relations:
    2*$.1 = 0
- m) b6 K! x% g4 b" }8 V( iMapping from: Abelian Group isomorphic to Z/2
/ e- ]5 F' E. q  P$ x" i5 ~Defined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M
4 q8 A) m- q8 U' l- m2 ]2
2
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
& i+ O& n" ~) J    2*$.1 = 0
! M7 B1 y3 Z8 ]1 QMapping from: Abelian Group isomorphic to Z/2
. Q8 [% h6 J+ z! u  nDefined on 1 generator
Relations:
8 I. I0 V* U6 U' |    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
2
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
" [' \. ]6 d' H# ?- W5 D! Z% F    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
# K, T2 y  b5 y0 Q' Z0 o; k* eRelations:
# ~- J8 B# a: R7 _6 t9 S( N    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
  A* S( p3 _/ k/ h' Rinverse]
false
false
==============


+ l& ^; y  D+ Z' d" eQ5:=QuadraticField(-50) ;
# S; A% c$ {+ K7 {7 q) F3 V/ KQ5;
' B9 x5 A7 Z9 |& i( I% F3 {2 R
+ `( i' P& s" {$ qQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
9 M! Y) \8 b) T4 @M:=MaximalOrder(Q5) ;
( `$ i- z/ {9 y& f3 s$ jM;
NumberField(M);
  X9 j/ B. U# @0 f- E) kS1:=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;
9 i! Y- ]  A7 lIsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+50);  
! r1 J5 Z0 z: sDiscriminant(Q5) ;
! P& g( ?( S) _3 G2 [FundamentalUnit(Q5) ;
- h5 W8 H/ a. i8 lFundamentalUnit(M);
Conductor(Q5) ;
8 b" q3 a1 \2 H- k1 {5 s1 U
% n4 r" T& g5 O; dName(M, -50);
0 l* }" D( w3 |! n- c+ X9 EConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
0 \( T( k/ p3 @; I7 E5 `ClassNumber(Q5) ;
& n. v# w5 `% B" j% b5 E, kClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

; `6 K6 s- d! l3 N, t9 `2 g, fQuadraticClassGroupTwoPart(Q5);
, I/ A. V% T% C0 z& UQuadraticClassGroupTwoPart(M);
NormEquation(Q5, -50) ;
" P- m1 G' h! U0 Y( X  q- _' ^NormEquation(M, -50) ;
' G, q3 y: h( w$ ]& ^: x% v
) ?/ K, G( i6 W) sQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
  S8 E; j/ e0 T2 FUnivariate Polynomial Ring in w over Q5
) t# \* e* F5 V- t8 ?Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Order of conductor 625888888 in Q5
& [! `+ W! o# {0 P; jtrue Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
" J8 a( P1 B8 d& D1 ktrue Maximal Equation Order of Q5
# e( R' |& j+ {; L& Z& K- Atrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
: u* M- |  N& d[
/ Z& j8 \: Y6 j- R; f$ C- \9 t    <w - 5*Q5.1, 1>,
7 S5 X) @8 G! ^# ?& e9 A* w    <w + 5*Q5.1, 1>
; }9 T) s! \1 J! K0 P4 B% \6 X]
-8
+ |1 @9 `& ?3 G: X0 {4 v
>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
% }! T' f- z) F2 Y
  }9 J  I4 D' f7 s/ M
. ~: K( e& v1 y* D8 Z+ G' M6 b>> FundamentalUnit(M);
                  ^
) L$ D0 U6 k" z) u, X' jRuntime error in 'FundamentalUnit': Field must have positive discriminant

) i# f+ }/ z% l! v3 F6 E/ U0 ]8

. P& N+ ]9 I2 d% d* a>> Name(M, -50);
3 j3 C- G, q2 Y! X; {       ^
Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]

& C9 F0 p# X) ?, N% `8 Z1
5 a) J5 ?/ m/ S5 y! DAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
/ x; R; f. X+ W: @, FAbelian 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
inverse]
1
Abelian Group of order 1
% v2 P, |! ~! x0 {: yMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
Abelian Group of order 1
- l1 {1 _; w: S8 ?/ {Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
6 X) i- F" ]! {9 n7 xfalse
false
* Y( e" d& m( o5 u% b

lilianjie

看看-1.-3的两种：

Q5:=QuadraticField(-1) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(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;
/ {5 h) a2 p- F$ h  X* [IsQuadratic(Q5);
8 L2 M9 Q4 C8 M3 r+ i# uIsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+1);  
- [5 @$ X! q% ]Discriminant(Q5) ;
0 O6 [' K& E. q) l0 u  @FundamentalUnit(Q5) ;
FundamentalUnit(M);
, ]+ M8 P) ?% A# ~) }- QConductor(Q5) ;
0 }$ N% g+ K7 u  f/ a1 z
Name(M, -1);
Conductor(M);
ClassGroup(Q5) ;
! d( ]% B0 ?* {, e1 y0 _( \& PClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
8 P4 w. j- v0 X4 D- OPicardGroup(M) ;
PicardNumber(M) ;

  M4 S# U+ \+ ]' tQuadraticClassGroupTwoPart(Q5);
; n# a+ P0 t" q4 CQuadraticClassGroupTwoPart(M);
- K2 J! w# v/ ]( w% L% t7 t& GNormEquation(Q5, -1) ;
NormEquation(M, -1) ;

Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
" @' o- {  n" V- v9 [) k$ IUnivariate Polynomial Ring in w over Q5
6 T; }* l2 m& r) q2 LEquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
( x* t& E; |& r5 U. zOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
% E- j& [" z' L" P7 f& e4 J3 x, ftrue Maximal Equation Order of Q5
8 ?: [4 v. c0 p" n% Itrue Order of conductor 1 in Q5
( x* ~$ y4 ^; F' [5 ltrue Order of conductor 1 in Q5
2 A' G7 @. Z' o- `& b6 ltrue Order of conductor 1 in Q5
) X2 l7 R( F( {% |- G6 O! q[
2 |3 H8 S6 e& G7 _; P& M    <w - Q5.1, 1>,
! C7 K# x' @( @1 k    <w + Q5.1, 1>
]
8 Z9 ]* L) n$ W2 H5 @5 d-4
/ g4 n: B, D9 u  _
* U* D* c$ K' e+ K' J; K>> FundamentalUnit(Q5) ;
                  ^
0 s5 n6 W! j3 P, T# bRuntime error in 'FundamentalUnit': Field must have positive discriminant
6 S8 c; P, _5 |7 F8 p* y4 \- f& L

( v& r7 B+ y+ i% U6 G6 z; Z* p9 {>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
3 e, U& L2 u5 q6 a0 c3 @2 [
4
9 X2 w2 n4 L! n* [* n6 T
( f7 y" i6 M: d+ I>> Name(M, -1);
       ^
& |. e- r! y, nRuntime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]
& `+ A8 p7 v7 ^: O& ]
1
% N7 T# `/ {! f0 {4 f+ VAbelian 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
3 H7 l8 _+ x  ]0 v- n. G" g# [$ d2 q1
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
( _. k$ q. M2 P9 G5 y2 b5 r( iinverse]
1
) w+ E3 w( A3 H6 LAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 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 i& R+ _5 t& n$ Q( V5 w-4 given by a rule [no inverse]
5 `2 j! Y" j" j/ S2 \false
false
===============
/ \1 P: r9 s  e3 L! G
) p) o. I! d) P3 m* EQ5:=QuadraticField(-3) ;
6 g- o9 V4 u  q3 ?9 {% y6 }Q5;
6 D0 d3 a  I! N5 u, ]  z+ K
, }  d$ H9 W" ?* \* y' \Q<w> :=PolynomialRing(Q5);Q;
! {4 e. a, \1 _* f; ?$ S6 SEquationOrder(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;
IsQuadratic(Q5);
' i0 v' N9 e% t: n# DIsQuadratic(S1);
: ]/ x. R1 Y5 F& XIsQuadratic(S4);
IsQuadratic(S25);
1 u- m0 S8 A; E0 C* wIsQuadratic(S625888888);
Factorization(w^2+3);  
' b. R# ~/ b2 L: |+ p' `Discriminant(Q5) ;
( e+ {6 }9 z) DFundamentalUnit(Q5) ;
FundamentalUnit(M);
6 D: G, m$ n. l9 K: KConductor(Q5) ;
" U& k6 ~% C* @/ X. d; z1 Q1 {
Name(M, -3);
9 f5 f" L' X4 I+ p  j" @3 vConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
$ i3 @7 H& P& m1 t' @% rClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

5 u2 H0 q' e8 l+ Q8 B9 U- mQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
  f/ s7 G) }4 GNormEquation(Q5, -3) ;
0 b4 b6 p$ z4 }NormEquation(M, -3) ;
2 x2 x1 E0 T& w% H
6 I, u. P5 w3 K" j. NQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
1 d, X, r% w+ c8 jUnivariate Polynomial Ring in w over Q5
! c1 E. Q  o1 |( O" d3 X/ r, pEquation Order of conductor 2 in Q5
0 u& N6 R3 W2 X. D; f; W: B; RMaximal 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
true Maximal Order of Q5
# u6 l( y3 L9 a- ^true Order of conductor 16 in Q5
+ t3 \# \( a0 ], m- A% ]. R/ Mtrue Order of conductor 625 in Q5
( r' c% w, X; c% E2 Ytrue Order of conductor 391736900121876544 in Q5
[
    <w - Q5.1, 1>,
7 }- a3 u# q$ m! n    <w + Q5.1, 1>
$ N. U; W! q4 y+ G5 K2 t]
-3
' h! S$ w/ L* Z% L; k, A) n% J: }7 {
; |( P4 U7 d* x% F>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
5 t9 ~2 ~: Y. T  O" _) w! a# i6 t

>> FundamentalUnit(M);
" D# z1 U' M5 u* n9 {                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

* `2 O- K0 R# S! ]3

>> Name(M, -3);
7 D% K7 ~8 d- U3 o" ?+ \" N; T% n       ^
" V( i6 k9 v* {Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]

1
Abelian Group of order 1
- h& t/ o8 z8 x. `& vMapping 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
: v+ I* @9 ~3 y) T1 b1
8 d2 Q0 f  x8 }3 r, c: [1 h/ gAbelian Group of order 1
& \9 ~$ M3 b0 YMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
( n# c4 B' m! l4 I$ ]inverse]
9 Y* y: L' U# ^+ M) a1
3 _% ~+ Q0 l( [- J6 ]% d4 f7 xAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
" a6 n$ Q: P) u9 G- J4 D; k7 k-3 given by a rule [no inverse]
3 ]7 J8 q( T; w1 }Abelian Group of order 1
( C: T/ R* R9 X, t0 T6 V' E( ~Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
! H% c) G( U/ H2 ?% Q, Y, B. H-3 given by a rule [no inverse]
1 u. s; L8 t' L1 n) Tfalse
+ W! o3 r9 j0 q4 {9 pfalse

孤寂冷逍遥

lilianjie

  f- a& x, H5 ^$ J$ X3 c0 m5 F
  z6 a' l- s) Z# pDirichlet character
( `: U  [% f0 aDirichlet 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

- E- D# K9 P4 [# k% ]# T5 C-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
8 C4 x1 f7 L( Z/ N6 |: t: R$ b
-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
) c  s  E1 E4 s2 J6 Y) A/ wh=-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，

! [) c4 N' t, X! g7 S" Q  N
8 g7 |* O5 v" w) k
1 j. L$ d# B% e( R# Bh=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2
1 V: B* w. C6 d6 Y! _0 m1 q9 B( @


-50时  个单位根                          N=200

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

8 ]) o; M) X" n( B1 V' f& ~3 }  Q
F := QuadraticField(NextPrime(5));
! ~9 v& w, {: P9 ]) B% O. w# H
KK := QuadraticField(7);KK;
. ]. \3 k3 F. o2 DK:=MaximalOrder(KK);
% T3 A; B. F$ [" IConductor(KK);
  |. D8 Y1 \, A' hClassGroup(KK) ;
* b+ x; L2 k6 A2 ~/ h' |QuadraticClassGroupTwoPart(KK) ;
+ y0 z9 h8 J* z) ?4 ?NormEquation(F, 7);
A:=K!7;A;
* W, H2 B. Y) QB:=K!14;B;
) E! }; [& d$ K8 y% MDiscriminant(KK)

% o5 {) R' E4 v5 S. N2 vQuadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
28
  J% _9 [7 X& ^Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of K
. |, z% r6 _. D* y* ^Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
4 P  q3 ^7 L6 o: f" O/ W    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
. S7 M$ ~. Z  o, qDefined on 1 generator
6 A& C6 _+ }+ U! L$ ?Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
6 z; p$ Q- d% b$ o5 ]6 W  Y6 e; o- Sinverse]
false
7
14
8 c" l& V2 a. z1 z. ^# x28

lilianjie

8 @, c; o4 q- `* {; U
9 R$ v; e$ U" Y4 W! _+ m; ?& D


8 x0 \/ b9 ~- N
3 D  c; K* e4 m8 X3 X9 n
+ ~7 `( H" {; }: m5 Y
分圆域：
6 q+ g# @* k3 T6 W9 XC:=CyclotomicField(5);C;
& p* M8 G+ ?, s0 h" fCyclotomicPolynomial(5);
C:=CyclotomicField(6);C;
CyclotomicPolynomial(6);
+ d& {6 N/ t' n% n: k5 u. hCC:=CyclotomicField(7);CC;
) \# Q+ [# X" jCyclotomicPolynomial(7);
MinimalField(CC!7) ;
MinimalField(CC!8) ;
; b7 \9 Q7 \( s' F. sMinimalField(CC!9) ;
MinimalCyclotomicField(CC!7) ;
- H$ K3 m( P' v3 z; _, v# g3 H2 zRootOfUnity(11);RootOfUnity(111);
* z) s+ u! C8 F+ r$ A( |3 mMinimise(CC!123);
Conductor(CC) ;
CyclotomicOrder(CC) ;

! ^  S: E/ k( e# _  B# iCyclotomicAutomorphismGroup(CC) ;

Cyclotomic Field of order 5 and degree 4
$.1^4 + $.1^3 + $.1^2 + $.1 + 1
Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
- w7 m, z& g0 D% D. kCyclotomic Field of order 7 and degree 6
$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
5 _/ a  a0 a* u% e+ [0 SRational Field
Rational Field
: j  d+ U( C; H+ hRational Field
  R1 I! V+ G, nRational Field
1 _- I2 P, z( w. V+ m8 ^7 l) ozeta_11
8 q3 G2 ^5 y2 qzeta_111
, X- p7 @0 a9 x$ ]8 x& i123
( o0 I* N' N7 Q# ~3 ?8 u4 G7
( C1 y* I) Y5 v) b$ P1 w7
  P1 O+ q7 j8 S- F. Q' EPermutation group acting on a set of cardinality 6
Order = 6 = 2 * 3
: W! b& h. f4 \& Y    (1, 2)(3, 5)(4, 6)
    (1, 3, 6, 2, 5, 4)
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
( \* r! Z, i% V% s1 A, j9 D* zCC
Composition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
; {7 c) O, K: Y. ~& U6 |1 IDegree 6, Order 2 * 3 and
7 \4 G$ m+ o7 K: x3 J* QMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC

lilianjie

' T  U0 w6 e' F" n

lilianjie 发表于 2012-1-9 20:44
5 m! w1 ]) N: X) K( Q4 x8 ~' L! x分圆域：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

分圆域：
4 c5 f7 k2 f- L* m' E, k% B分圆域：123
- B. F0 i1 `+ c5 v: W2 I
1 y1 Y4 J9 n; [. sR.<x> = Q[]
F8 = factor(x^8 - 1)
F8

(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
1 r" t1 {$ D. q
& n( h) g! N+ [& o& @* a" FQ<x> := QuadraticField(8);Q;
+ f: l6 z+ p* D4 J3 g. jC:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;
, v( q& t( U  E7 j
F := QuadraticField(8);
F;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
1 ?1 }- w1 |& |8 l3 M' @( _7 Y+ RCyclotomic Field of order 8 and degree 4
7 b" A, E0 `1 w( _$.1^4 + 1
+ U& l4 d0 y2 Q+ b4 j0 d0 lQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
! e( H  b) V/ Y/ L4 I4 G[
* N+ w# U7 S8 ^, `8 m! a    <$.1^4 + 1, 1>
1 ^* d, a# p6 V" R: U+ V]
# q- Q1 i( L" Q
R.<x> = QQ[]
- q* J, r& g; ]6 ^* r+ tF6 = factor(x^6 - 1)
F6

(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 b( g' a/ p9 g
Q<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
  m6 L7 ^; ~. Q: p2 TFF:=CyclotomicPolynomial(6);FF;
. x5 ?( o4 X5 U$ t
) X) r; I) T) |1 h  t( iF := QuadraticField(6);
: r. c4 k+ K. o/ kF;
$ _+ T# k$ r9 ?6 S+ L2 tD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
% R; O% k+ b) Q  nCyclotomic Field of order 6 and degree 2
* y2 [! @2 \9 `( [5 r4 ]$.1^2 - $.1 + 1
$ a7 Z% X( K/ M3 l, LQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
# T/ P. @% c& g% Y& H, N' _# I  w    <$.1^2 - $.1 + 1, 1>
]
2 I' g7 z' [& \; T
R.<x> = QQ[]
+ M4 U  j4 u5 TF5 = factor(x^10 - 1)
$ S, q! @; x2 o3 d( aF5
9 I0 V. N: u, V7 u(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
- q. {4 |' V; m9 Y1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
" z* ]  C* Z# Y. L8 j' F
3 D$ x9 {( b+ Q- QQ<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
$ w0 x) m% `& Y" {5 l+ OFF:=CyclotomicPolynomial(10);FF;
5 ?% Q+ I3 }: o& q* T! Q1 p
F := QuadraticField(10);
F;
D:=Factorization(FF) ;D;
' w+ ~% Y9 Z6 P* uQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
: q* e. r' F: T* U1 ?7 H7 E7 a9 e$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
9 Y6 ^9 u% f8 Q2 k6 v, O4 Z    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

lilianjie1

分圆域：123