虚二次域例两（-5/50）

lilianjie

/ H: N$ J1 p: sQ5:=QuadraticField(-5) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
* t7 R' j" j( V$ m8 ~. k' f0 rEquationOrder(Q5);
: s6 ~# o% o3 m. VM:=MaximalOrder(Q5) ;
& F/ T+ y7 S) F; RM;
NumberField(M);
3 T+ A% B9 D- }+ P+ g7 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);
IsQuadratic(S4);
. `2 |- a* E2 H3 _0 HIsQuadratic(S25);
IsQuadratic(S625888888);
+ @& m2 t; G7 |2 M9 O) @; ?Factorization(w^2+5);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
5 R  l) ~% [, Z4 _; @% EFundamentalUnit(M);
Conductor(Q5) ;
; X( {: |# M1 W; F) s( h- {
# ]8 ~: t4 m# o5 _Name(M, -5);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
1 B  O# X" f" d9 Y6 FClassNumber(M) ;
; n: C! V; ?- T/ h7 Y; a3 W* O( CPicardGroup(M) ;
- V: p1 P" @, z: VPicardNumber(M) ;
! S, Z1 n) t  N' }7 O
$ h, N0 Y8 M; J2 pQuadraticClassGroupTwoPart(Q5);
& }- M0 Z" `& m* ?3 K# XQuadraticClassGroupTwoPart(M);
' [/ N  Q7 Y7 a/ F0 H) i' V& hNormEquation(Q5, -5) ;
NormEquation(M, -5) ;
5 e( B- O4 z+ o/ ]; }2 x0 tQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
# L" O5 \- a  Z7 j- W; c8 h  |$ N; `, IQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Order of conductor 625888888 in Q5
8 [3 |; _  T# Y) Itrue Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
$ P" M: e) W% ]true Maximal Equation Order of Q5
! i, P3 f+ y# \  Ztrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
( ], t+ {) _. S$ i1 `[
! C9 O8 u: k2 Q  g# _) S    <w - Q5.1, 1>,
    <w + Q5.1, 1>
+ X8 w7 t2 s9 I, f% ?; y6 k% r]
' Y4 [( a' ]: C5 F% l4 W. p-20
6 }1 P- d/ o: ]" F# S$ ~2 U
: D, o) Y- c+ D0 J>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
( m) K4 o* j" G0 B5 _6 \! {- o* N

8 @1 O# n( i" K  |9 h; X>> FundamentalUnit(M);
4 [7 ]" Z$ t( F% N6 W+ q                  ^
' M# W1 X! P0 a4 m0 M0 o8 iRuntime error in 'FundamentalUnit': Field must have positive discriminant

20
7 z! O$ T( o6 M' e6 r
>> Name(M, -5);
       ^
Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]
$ N, o$ o1 o; |# y
1
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 Set of ideals of M
6 R$ C" D. a5 S, v+ Q' A; r9 ~! DAbelian Group isomorphic to Z/2
# m, ~$ T* ?! F! i3 r" T) W& L( k4 |/ ~Defined on 1 generator
  E7 F8 p$ c0 {/ |: [/ nRelations:
) n& U- U! m3 ~- f; |5 b9 x    2*$.1 = 0
* o! H) A9 a, _Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M
2
2
/ a4 E: M% O4 k* CAbelian Group isomorphic to Z/2
6 z0 t' O5 k  a  g% P* V# H; X" ]3 DDefined on 1 generator
/ {" @' y& k( E6 _; _0 O. kRelations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
' G! ]. z- N& F7 e3 W6 r& xDefined on 1 generator
" V$ ?+ o( t. @, HRelations:
    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
5 y- @7 A) O% w5 S3 _2
. r  ~& u5 a! H5 H. E' uAbelian Group isomorphic to Z/2
Defined on 1 generator
# I/ F1 K, D) j% v" TRelations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
; K+ v' S& U3 t/ tDefined on 1 generator
9 ~+ W8 m5 |- T5 L8 cRelations:
- N9 ~$ n2 @; |/ J# b! a* k    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
8 y4 ]* g! r! S, a) rAbelian Group isomorphic to Z/2
Defined on 1 generator
2 M2 G/ R- h$ iRelations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
3 ?7 w+ n- D/ ~9 q: L9 m4 s0 NDefined on 1 generator
Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
2 O. q, a) @+ o1 }false
false
==============


Q5:=QuadraticField(-50) ;
7 q) k0 j- t/ R# wQ5;
6 h) g9 x3 d5 b
Q<w> :=PolynomialRing(Q5);Q;
9 A8 V# R( F* @  {3 ?EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
9 w% @( v& K1 T& j' b! t. w- w  KM;
NumberField(M);
! H# R+ n% ~) R& eS1:=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);
& q. X. j  ]. t8 S3 f* {+ HIsQuadratic(S625888888);
5 ~9 ^, b9 `4 ^5 Y- JFactorization(w^2+50);  
# W0 X8 ?. r* w9 U$ ^8 V. u4 S& qDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
: c2 {; T  l. t% }+ q; ZFundamentalUnit(M);
Conductor(Q5) ;

Name(M, -50);
6 M; {1 e  o( e1 q. W" nConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
) ^, n; z+ T0 I: k( {/ d3 yClassNumber(M) ;
PicardGroup(M) ;
0 a  W6 y2 M. k( ]. s7 q( V+ bPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
3 L2 ~" g1 p# }. Q/ uQuadraticClassGroupTwoPart(M);
NormEquation(Q5, -50) ;
: T" J) d8 d) |: m& aNormEquation(M, -50) ;

Quadratic 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
6 E8 d; O, w) `0 G" @Maximal Equation Order of Q5
* _, c5 v. w+ }* w; {' p5 o0 ?Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
( K3 u+ ~+ a2 e  i' o3 [Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
' k: Z8 ^/ ~7 g9 Q+ {0 strue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
7 q- K8 t, L! |3 b% l- @true Order of conductor 1 in Q5
7 K3 Q6 U# b. v5 M3 H3 M. otrue Order of conductor 1 in Q5
; u5 ~0 Y: r7 W) o9 M[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
2 \  h* h+ U+ E3 P4 Y* {* \4 f]
! j7 _1 H( `; t  ?/ `7 l-8

>> FundamentalUnit(Q5) ;
% c) F7 U2 S& I' e8 ~1 P6 L                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant


+ D4 P6 o; r! S+ K' ]1 a  O# O>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
$ S' `1 t# E: H1 {
8
$ b' O/ U( A$ v) m4 n' Y
- Q9 S" o" S3 h  C>> Name(M, -50);
( r/ D# f& }* }  a4 @# y5 Q8 h; o       ^
4 N9 m. q$ s; N' n) VRuntime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]

$ U/ i7 m6 v0 c1
; m! t1 Y  ^( H+ n7 EAbelian Group of order 1
, k) ]5 G; N7 }0 vMapping from: Abelian Group of order 1 to Set of ideals of M
; r" a" T3 a9 {( V! @1 LAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
2 B; w: l1 ~( e* T7 G! d1
2 m! B1 n" |3 h4 F  R, B4 D/ @" eAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
  n# c+ K- N8 K- i/ @/ \5 v- zinverse]
- F4 Z" c  q; `8 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]
& r5 N. I2 ^' [false
" B" @: r) J5 v* o4 jfalse

lilianjie

看看-1.-3的两种：

Q5:=QuadraticField(-1) ;
Q5;
# }; x, h0 u4 b9 U
7 b. u6 Y2 S& |3 M& FQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
. v4 {9 R1 r6 [. e2 U( C  D0 sM:=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);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+1);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
- G) V1 N1 q; r9 U+ y2 m8 ^0 wFundamentalUnit(M);
Conductor(Q5) ;
& J2 d# `5 i5 n9 X* l6 C$ y; ]
Name(M, -1);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
5 G2 r5 Q  v5 F7 h& K( l& BClassNumber(M) ;
. ~5 r7 U: v- a+ T. a! T8 u0 w9 mPicardGroup(M) ;
PicardNumber(M) ;

5 _2 c: d. v" Y& e6 `QuadraticClassGroupTwoPart(Q5);
, ?5 V; F, |/ bQuadraticClassGroupTwoPart(M);
+ x3 Z, U8 H/ @+ BNormEquation(Q5, -1) ;
8 J9 M6 ~, Y! ~5 V( B9 L9 c* JNormEquation(M, -1) ;

Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
, G; V, a  z  m4 C/ u' hUnivariate 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
0 z+ a9 {0 u& k0 B& V( HOrder of conductor 625888888 in Q5
; _7 ]1 S0 p. E5 z0 T! Y# \$ _* z7 ftrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
+ N) G) V* N# J8 |. `; ]0 C$ atrue Maximal Equation Order of Q5
# S; ]6 M4 V+ D9 W) {true Order of conductor 1 in Q5
8 t  `/ i& e6 @+ `true Order of conductor 1 in Q5
7 [9 i2 I( h) d  @% ntrue Order of conductor 1 in Q5
- ]6 E* @7 ~% ~8 X# o  P[
* N1 u2 X( a* g) \' t    <w - Q5.1, 1>,
    <w + Q5.1, 1>
$ Y0 o8 P( T: o! u/ W) g: N. A]
3 M* B: @/ }6 h, I! f-4

1 m, w+ R; t4 V% y- X9 I>> FundamentalUnit(Q5) ;
                  ^
, d1 b0 d. P! y1 `0 r4 U- {! PRuntime error in 'FundamentalUnit': Field must have positive discriminant
* f8 G7 x8 l1 K/ v: F& L6 F. _& [

3 `5 {+ ?6 n$ V2 a>> FundamentalUnit(M);
/ f  `* @* s. \' r& G; p                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
; M, J- w7 L6 a) T* S% I6 a" j8 a
4
$ e; o/ R- x4 [, f7 w' R( o4 R
>> Name(M, -1);
       ^
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]
5 W6 b5 E7 X, Z3 l
! f% y/ k# R- s" t8 D0 l9 |" w1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
: y" o# C# w4 f7 b6 P* F' i* dAbelian Group of order 1
. k) ], ~- O) ?, @" N; N8 NMapping from: Abelian Group of order 1 to Set of ideals of M
( K& F+ t4 S7 e' x% z/ q6 X, e& y; g1
, E: D  O: C( u; z# ?1
/ V; A& B9 R$ y; m; PAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
2 \, W! V# Q) E) g. l9 G) ?7 d  u1
; @* z- _3 `/ c! bAbelian 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
+ T1 L; P1 V. C* y: ZMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 given by a rule [no inverse]
; b' ^  K' V% Z* Z# O$ f' Efalse
7 m9 C7 s+ n* U$ U5 b% R# ^0 pfalse
% c% Y1 T% }9 V7 {+ q, l===============
) y. o7 e$ h: W7 A/ X; c" Z
% l( ~5 q" j( a" q4 v) _! ?  t1 zQ5:=QuadraticField(-3) ;
" ?# b& z3 A8 a" R6 d$ z8 W' C5 o; j9 |Q5;

9 U( y9 y  T% \4 P  ^8 l: ^6 VQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
! J0 T1 S4 k3 K6 v+ DM:=MaximalOrder(Q5) ;
M;
+ n4 ^9 ]1 m8 aNumberField(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);
: |; V7 F' V" M" i  e8 O; m- F) xIsQuadratic(S4);
IsQuadratic(S25);
4 q" C# e2 q% X7 G% `IsQuadratic(S625888888);
$ {+ |. K8 L8 @1 d2 Q2 w" H, zFactorization(w^2+3);  
Discriminant(Q5) ;
( c; p$ c  P) Q2 C$ g6 ]- M3 a4 tFundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

Name(M, -3);
Conductor(M);
! U' b9 V1 X" TClassGroup(Q5) ;
ClassGroup(M);
# D5 R- @8 i2 Q9 X& S5 yClassNumber(Q5) ;
ClassNumber(M) ;
$ N& f* r% M& z* h* G5 W& ZPicardGroup(M) ;
* ?+ x& x* |. t) QPicardNumber(M) ;

0 H, N6 M8 N1 j2 U: R$ n8 f) JQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -3) ;
: p4 C0 m- e) C# z; o2 k0 TNormEquation(M, -3) ;
6 [) ]  `7 V  e$ k# \, |2 \
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Univariate Polynomial Ring in w over Q5
: l/ |$ w5 G# t& g/ D9 cEquation Order of conductor 2 in Q5
Maximal Order of Q5
2 I5 ?+ i4 o# k; |: _& ~4 ~* H9 |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
: B7 I3 U" r8 @3 b  W6 }0 [6 Strue Order of conductor 16 in Q5
2 c# y3 R1 h6 `. q7 j5 p- M: Ctrue Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
- A' B* d& f3 N9 \( g]
, v: `% e3 ^: G+ J7 e2 Q  D4 v-3
7 c6 j8 R+ ~$ ^& n+ L- A
- E  Z+ K1 H! E( D+ {>> FundamentalUnit(Q5) ;
                  ^
& P! p2 V! [9 r9 f$ XRuntime error in 'FundamentalUnit': Field must have positive discriminant


>> FundamentalUnit(M);
' u, i. d& F4 ]5 S7 W3 @3 {                  ^
8 L# d2 p/ I$ l' ^+ `  aRuntime error in 'FundamentalUnit': Field must have positive discriminant
4 u9 N! a5 N1 a1 n
& O. S( A4 x8 P: W, c3

>> Name(M, -3);
       ^
: t% s/ t; M/ P! f7 ^Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
0 W9 F4 U% c1 V
1
( c! C& D! s& fAbelian Group of order 1
; Z$ K' r0 `! j* n% p$ K3 wMapping 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
0 Y6 O7 C6 [7 J, L& ^1
Abelian Group of order 1
% E7 [5 C( G: z1 h9 d" D7 |* pMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
, B! R1 _5 G* p2 E' }1 }1
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]
3 a5 m& P: h1 TAbelian Group of order 1
# Q5 O3 v) ?. z7 f/ a, }- H8 ?; yMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule [no inverse]
false
: |% y% Z, F' E; r' @) G( f) |5 ]! tfalse

孤寂冷逍遥

lilianjie

/ x0 u" H0 o6 Q8 M5 F
Dirichlet character
Dirichlet class number formula
; t& m3 G9 c& N1 z5 N
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
% O* ?& }; v& B7 {4 P$ ^: }1 `" t
-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

- w  Z5 _3 {! q. G0 R* ^. q-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，
6 m' r4 p% ~. z! j* |( ^


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

; e1 ~) |& a# @8 j. Y
. q" R  @0 L7 x0 H7 u# M0 A
7 I/ `( u& o7 K( W% K# R. X' [-50时  个单位根                          N=200
3 z% Z, D6 r4 x/ r3 ^

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

' ^( V9 f+ R7 {, Z1 o5 S
F := QuadraticField(NextPrime(5));
+ o3 R/ f% I1 K0 D
KK := QuadraticField(7);KK;
K:=MaximalOrder(KK);
Conductor(KK);
) w6 r9 O" r% W- o& T1 iClassGroup(KK) ;
QuadraticClassGroupTwoPart(KK) ;
4 S1 f# K* W% V2 D( |) VNormEquation(F, 7);
A:=K!7;A;
: H: U$ \1 H, g2 D4 o$ LB:=K!14;B;
( S& d% o1 w% e5 Y2 v% x' ^3 oDiscriminant(KK)

Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
5 d5 z- c' g3 s0 M9 B( g28
  `3 M$ `: t. r+ o* O, SAbelian 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
" o- v* p; u( }# zRelations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
+ j  ]% |) y) H" G9 q# l/ k8 \    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
2 |, w% D+ g6 R% ninverse]
4 d7 _/ u! U! d% `- Y4 ~false
7
8 i8 r* ]7 A) }- ]14
7 P3 V5 h. D; }; u28

lilianjie

  v' F5 w5 n5 c# t1 @1 t
- g( C3 _. [/ y
* D0 r2 u5 s- N: m
' m7 n5 H, j6 a1 u  K! {$ y0 D1 F
# x8 c- {# \) m, h) W
$ a3 b* @3 b# @. C, v分圆域：
C:=CyclotomicField(5);C;
: B6 X3 U! w$ H% y; J# a, [2 ]CyclotomicPolynomial(5);
, l8 v9 [5 Y6 v; hC:=CyclotomicField(6);C;
CyclotomicPolynomial(6);
7 T( \6 X/ k/ C2 [9 J/ ACC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
MinimalField(CC!7) ;
MinimalField(CC!8) ;
MinimalField(CC!9) ;
: I# q* U. {7 k( i5 nMinimalCyclotomicField(CC!7) ;
RootOfUnity(11);RootOfUnity(111);
Minimise(CC!123);
Conductor(CC) ;
CyclotomicOrder(CC) ;

CyclotomicAutomorphismGroup(CC) ;
, R. O" F' H. L+ u2 B
Cyclotomic Field of order 5 and degree 4
$.1^4 + $.1^3 + $.1^2 + $.1 + 1
1 Q1 H6 S. [7 m* aCyclotomic Field of order 6 and degree 2
* o' g7 m6 H$ N( A. M! C$.1^2 - $.1 + 1
( j- M8 O$ `: X* p+ `# `Cyclotomic Field of order 7 and degree 6
$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
Rational Field
Rational Field
& f4 l! J) K  l+ RRational Field
zeta_11
zeta_111
, L- {  d: I& `0 j# J% I5 n& N$ C123
( ~0 F; c6 i" S6 a! p: _% b7
' O# t$ d' b. F) L; R' U7
Permutation group acting on a set of cardinality 6
! J2 W. Z* n: _: G2 x* nOrder = 6 = 2 * 3
) D7 }" L* K. f) P" H4 x/ u    (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
CC
: K, A0 h% I, p' Q" d1 uComposition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
1 A6 v; i4 X/ q6 E# gDegree 6, Order 2 * 3 and
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
( ~$ U7 G4 ~5 {; ^4 [4 S+ V  MCC

lilianjie

" x1 X/ C$ J( u, q

lilianjie 发表于 2012-1-9 20:44
分圆域：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

% S! m8 o, M3 ^分圆域：
0 h. F: c0 a: {3 A分圆域：123
1 O9 w/ I% `1 P( [
R.<x> = Q[]
9 H5 m. U( l4 ^- _2 l2 w9 K4 yF8 = factor(x^8 - 1)
# o1 O2 Y8 Q& _6 n5 X' u, xF8
( v) I: T8 C: t3 M/ C9 L0 e+ L
& a1 b/ X. y" I  E(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

. s" Y" U( q4 d7 \* HQ<x> := QuadraticField(8);Q;
/ ]1 ]5 v% d6 y) m6 nC:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;
, K) j  j& ?  ^
F := QuadraticField(8);
" [+ K2 h3 W/ T& K( {% cF;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
! H& ^" ^$ D  W2 I5 T4 C, Q8 KCyclotomic Field of order 8 and degree 4
( h: K# Y$ d- {, [$.1^4 + 1
/ U, D& \- s; D# N; o0 |7 [) TQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
6 F" C* F2 p; y4 o( ?& I[
    <$.1^4 + 1, 1>
, E/ J3 _* H8 W]

R.<x> = QQ[]
F6 = factor(x^6 - 1)
F6

* c* ]. ^6 U: N& b2 }1 g, N(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

0 S9 R) A! M5 d+ P# bQ<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
1 F* a( w8 x5 D3 H# rFF:=CyclotomicPolynomial(6);FF;

+ _8 o+ `* F  x1 t7 a4 DF := QuadraticField(6);
" o% T( Z) w$ p" b1 z2 VF;
* t7 O3 l% s: [D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
6 N( Q! J6 r( j& k; P6 f) n6 Y, V4 l$.1^2 - $.1 + 1
* O% F  E) M: p, fQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
0 o5 T  c: b7 S8 e+ f& [[
    <$.1^2 - $.1 + 1, 1>
( L* e. e- {7 C]

R.<x> = QQ[]
F5 = factor(x^10 - 1)
F5
(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)
' y5 ?8 d$ V# d/ {* H
Q<x> := QuadraticField(10);Q;
6 R- l* A9 F" p% Y3 {& kC:=CyclotomicField(10);C;
7 ]$ f3 }' `. c8 S' u2 rFF:=CyclotomicPolynomial(10);FF;

3 E" `& a- H4 iF := QuadraticField(10);
F;
D:=Factorization(FF) ;D;
5 U9 |3 ^& z9 F! W; B" TQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
  o* ~4 D2 I# [: K3 v+ k4 d/ w$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
1 g- ]  q( v+ T1 N/ S[
+ G# E& N5 E& J( {3 r/ I    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

lilianjie1

分圆域：123