虚二次域例两（-5/50）

lilianjie

% t! K" L. A- {3 k# j
Q5:=QuadraticField(-5) ;
9 i! z5 z4 c& K& {: MQ5;
7 L! N, ]; ?) {/ x* w! P  V1 d" D
Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
4 c) F  z0 S9 T6 a  \$ OM:=MaximalOrder(Q5) ;
  }" x' K: u' h$ t  s; }' q8 ?+ tM;
* y9 s' N& i" @8 k+ v' g2 j4 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;
$ o9 J( j! ?/ cIsQuadratic(Q5);
IsQuadratic(S1);
' a5 X& z( ], k" ]8 @" O! CIsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+5);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
8 t2 x8 \* s: _9 A( w0 fFundamentalUnit(M);
Conductor(Q5) ;
' H, }4 V) f  e" N
7 {9 |& z& S" I9 [" ^! J$ VName(M, -5);
Conductor(M);
ClassGroup(Q5) ;
% ~) @3 _8 v& z: m1 {ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
8 [$ S0 [6 J' P3 |: r  n/ kPicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
5 `4 E5 Y; ?0 P* ?NormEquation(Q5, -5) ;
NormEquation(M, -5) ;
7 P5 E$ W- J: SQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
5 s% a2 G: q& LUnivariate Polynomial Ring in w over Q5
: E2 u" ]3 p( ~/ f  Q& U4 xEquation Order of conductor 1 in Q5
- `% T7 ?3 i( \- z5 |- nMaximal Equation Order of Q5
$ j1 ?4 |. u+ DQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
8 V% c6 f) _$ x) R+ I$ Z2 EOrder of conductor 625888888 in Q5
" b: {- v4 E) Y3 e( ~2 v$ @8 Q5 L& Otrue Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
  n* @9 p, a$ c6 S% A+ Ytrue Order of conductor 1 in Q5
  F( ]  u# Z  W- X0 G$ _true Order of conductor 1 in Q5
[
8 h+ ]9 H) V# Z, v0 E- l. K6 Q    <w - Q5.1, 1>,
7 S$ R( p6 G3 \. a    <w + Q5.1, 1>
]
-20
- Z0 f+ e0 l3 Y! V/ [3 y& I
2 d, F. A$ D9 r5 [2 ^# P>> FundamentalUnit(Q5) ;
' y8 }, U& t6 _- s                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

/ S" u: I0 J+ J5 r- K0 [& z
>> FundamentalUnit(M);
8 X  V3 R6 N! \                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

! p6 T5 w) V+ R6 U# d& h; @) t. U20
% W2 i/ T2 M! G. h
>> Name(M, -5);
       ^
& V4 i) J7 z- h5 H5 M- d8 kRuntime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]
: _3 R- ^5 l, q* v; D/ l
' c& z3 k5 C& @" w7 p1
Abelian Group isomorphic to Z/2
; O; t2 a4 ?9 R  o* ?) c+ [" _1 SDefined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
% s8 _# c$ y* HDefined on 1 generator
& O( M- j9 S+ z4 |Relations:
/ M, Z" ~: f' M- j    2*$.1 = 0 to Set of ideals of M
3 |4 F$ O- g; e5 e! n! N- N2 MAbelian Group isomorphic to Z/2
Defined on 1 generator
; X1 x4 J- O$ T1 i  g" X. DRelations:
: l9 M$ _7 d4 j* O5 j9 {0 h; s6 }    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
9 ?* y& K; K1 z% F$ a' YDefined on 1 generator
  ~5 ?& U& d+ M# _- y, F& VRelations:
1 `+ ?+ l4 i! e/ U/ w7 i    2*$.1 = 0 to Set of ideals of M
2
+ q+ I$ m/ x% h. E' D, g2
Abelian Group isomorphic to Z/2
Defined on 1 generator
& J$ K, Y8 B7 c. _) n/ c) dRelations:
    2*$.1 = 0
( \( Q( O8 M- J0 X$ W' T  ^Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
6 B. u. ^5 T3 G5 c" {2
! ~1 N/ w+ E' d9 {, CAbelian Group isomorphic to Z/2
Defined on 1 generator
) B5 j) p6 T0 H8 [Relations:
' V1 }8 s: }. {4 p, e    2*$.1 = 0
, Q" C2 m/ o8 _, vMapping from: Abelian Group isomorphic to Z/2
- \* q) c9 D' c  J% pDefined on 1 generator
5 v3 h. U# ~! Y3 e0 O+ p$ L! C1 KRelations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
9 w5 m9 k9 T4 L5 k0 a: ninverse]
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
* H; y  g3 Y2 ]9 _5 v8 ^! hMapping from: Abelian Group isomorphic to Z/2
% x' f1 F; D  rDefined on 1 generator
Relations:
# Y% F! T- h1 K3 l' L- l( ]% L    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
* i* L8 q9 L# L4 G) x% iinverse]
false
false
1 Y/ c3 g: R8 P& C* w+ P- s/ {" C==============

0 a$ J$ m0 ~1 j# A# }( f- {
+ Q7 B4 _9 i( `% p; T7 EQ5:=QuadraticField(-50) ;
Q5;

2 T0 W! i- \/ t1 d- }Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
- O/ t% W) a1 Y1 uM:=MaximalOrder(Q5) ;
M;
NumberField(M);
! Q) G( p2 `5 Z0 ^) i; Y& LS1:=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;
) q$ E, ^5 J& MIsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
; ~/ E2 @: Y7 S; N( YIsQuadratic(S25);
IsQuadratic(S625888888);
4 l. V( j& t8 ~7 w  PFactorization(w^2+50);  
% g, m5 j) Y! Q( G4 @, @Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
* V* J  L+ p" h
Name(M, -50);
Conductor(M);
% }6 `5 w) x2 u+ s8 v# ?( dClassGroup(Q5) ;
ClassGroup(M);
% Y/ n* B) I* I+ L1 q6 LClassNumber(Q5) ;
ClassNumber(M) ;
" }$ z/ x3 p9 lPicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
9 z$ C) x9 f$ }& f6 c/ GQuadraticClassGroupTwoPart(M);
* m5 K/ X/ L/ Y5 oNormEquation(Q5, -50) ;
# E7 }3 D  T5 u$ ENormEquation(M, -50) ;

Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
- K9 Y  N" u( g  \# ]! GUnivariate 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
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
. b, X& c! w# c( y+ Xtrue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
+ S8 K8 i& n2 i+ Ktrue Order of conductor 1 in Q5
7 W9 J, F4 j! z% e& e# v3 D1 [[
    <w - 5*Q5.1, 1>,
, g3 w- ~* {- ?; n% a    <w + 5*Q5.1, 1>
1 a3 T3 q& f. C0 h: d]
-8

>> FundamentalUnit(Q5) ;
                  ^
9 H/ [$ X$ C& M3 MRuntime error in 'FundamentalUnit': Field must have positive discriminant
, b+ Y; S" i) d6 [

6 |/ O/ y- n7 {# f>> FundamentalUnit(M);
6 R4 k. R' j- v5 E1 _# ~                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
; Z* B/ D* U. d4 H$ l3 K% j
: w- c3 i3 S* x" W8
$ i5 b/ l. W, p
- p5 t  t# f# q9 E>> Name(M, -50);
       ^
! I$ h! j0 I2 L8 TRuntime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]
7 Q# v  b# Q% g" p
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
7 S) N; m3 @) z. [/ p; vAbelian Group of order 1
7 j+ T+ n; G" L; LMapping from: Abelian Group of order 1 to Set of ideals of M
1
) X4 ~( K; r  N; l- T' X7 b1
; q& T( C& m! X* D) V) vAbelian Group of order 1
( h, t5 f# Y6 N! [  g! ^Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
- R% Q+ l$ r5 O0 z) ^1
! U( H2 w8 _8 H3 J) F1 t7 OAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
9 U! R5 J+ D1 o) w) N6 k5 XAbelian Group of order 1
" L& X! C2 d( W7 T4 U0 j2 u5 LMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
" a5 k, O. q# V- Q-8 given by a rule [no inverse]
false
false
: T0 q4 U5 P3 S

lilianjie

看看-1.-3的两种：

- P6 A1 K" e" r8 ^- G" `Q5:=QuadraticField(-1) ;
( _/ b7 @  |  @: `; _Q5;

Q<w> :=PolynomialRing(Q5);Q;
" K) S- `1 z; Z; A3 Z# q4 U8 TEquationOrder(Q5);
: z! Y( M% B2 o$ W. l" p9 t4 oM:=MaximalOrder(Q5) ;
# j0 z* e5 M% b) L6 v6 }M;
NumberField(M);
1 X; K; [+ K6 i0 |6 @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;
6 \* a- _; a2 e- _IsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+1);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
& j' M8 C+ u  `) }8 RConductor(Q5) ;
: d  @3 c  `) n) U
6 K. m! {" n) G# H% l! R/ jName(M, -1);
Conductor(M);
+ z% m1 c; Z% {: F7 @ClassGroup(Q5) ;
4 p& K- B$ e7 `* YClassGroup(M);
/ {" _: g& z% o. I& WClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
8 c; m. v* Q( @" pPicardNumber(M) ;
; c2 J' D/ x* ]4 e, a  H
/ L2 t; h3 I* mQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -1) ;
NormEquation(M, -1) ;
9 n- d7 o: ~' C5 @! T+ m7 o
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
/ P6 l& f) _4 o2 J. eUnivariate 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
Order of conductor 625888888 in Q5
9 X% Z+ |1 U0 }  q: Ztrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
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
[
& H* O- [6 z* n8 M" {    <w - Q5.1, 1>,
    <w + Q5.1, 1>
5 g% L) [) r* R2 \& I0 k]
0 C* m+ f- d* h( F-4
, |1 {0 s7 \. o' ^1 j
>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

1 k' @2 w, A# j: {. {! c4 ^1 a/ K7 a9 Z6 E
>> FundamentalUnit(M);
% h; }" O5 _5 E                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

& Z9 u8 g$ @1 k4

>> Name(M, -1);
       ^
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]

1
) U$ p6 ^9 C/ Q- @/ sAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
( s, r$ u0 z6 L, }* I/ v4 U" pAbelian Group of order 1
' U4 ~6 o& @- n3 f) L1 s3 LMapping from: Abelian Group of order 1 to Set of ideals of M
1
& t# c( K! \7 [( E1
Abelian Group of order 1
9 L; c/ p- @5 k: ]& G8 wMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
Abelian Group of order 1
# V4 \! R+ w( OMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 given by a rule [no inverse]
) G3 D) c+ V  W0 V! B4 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]
false
false
5 S9 Q1 S) _5 E+ T' o===============
. f1 X* o( V' D, n5 O& L, z& `) S
Q5:=QuadraticField(-3) ;
- r, O+ }4 b% X+ o' hQ5;
! B, b! i; h) x. u1 ]# f( ~0 o9 J
Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
6 C# A4 U) C* ]M:=MaximalOrder(Q5) ;
7 L+ T4 n. Z* R5 o+ S; o% DM;
4 N: n6 J+ [8 E/ w" |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);
  i: v9 I4 d' \0 T  q2 h: fIsQuadratic(S1);
IsQuadratic(S4);
( T/ E& ?" j$ q: M5 Q3 q) hIsQuadratic(S25);
IsQuadratic(S625888888);
. N  M  A9 x9 W) s8 }5 _7 eFactorization(w^2+3);  
( I. ^: I; \" R% `* }4 ?) ]) YDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

Name(M, -3);
+ x# L, {. V8 |5 x/ [6 t' yConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
4 T/ Y+ T5 j) N. ~6 Y. r4 `+ s% [ClassNumber(M) ;
- o$ [8 U6 |5 ~8 H& MPicardGroup(M) ;
/ n" j0 ]) S# i+ J8 [6 @  [PicardNumber(M) ;

8 \: X4 ]- I0 I8 l" GQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
7 _4 J; h: K" ]7 q# _. J$ aNormEquation(Q5, -3) ;
NormEquation(M, -3) ;
. x* l) V4 M" b* q- p$ L
& ^/ r) X- ~, Q3 l3 d7 c9 w/ T' v  VQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Univariate Polynomial Ring in w over Q5
! C1 O5 O' H& n  \! m! g8 a. ^5 h4 fEquation Order of conductor 2 in Q5
  ^& k3 }% k: \' g8 ]2 LMaximal Order of Q5
2 m& V2 |6 e' r8 N8 _Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
+ z3 p+ h' [% AOrder of conductor 625888888 in Q5
% p9 o4 Q, r& J5 O1 @true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
true Maximal Order of Q5
4 E7 O) |+ l9 y2 V' I  G1 ytrue Order of conductor 16 in Q5
3 J6 H8 s9 I4 L) D. Dtrue Order of conductor 625 in Q5
9 w! Y5 b& _( M7 E, j* W6 atrue Order of conductor 391736900121876544 in Q5
[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
4 ~& v. l) W8 _4 V" ~]
; ^" `4 q3 E; F-3

) m; c# O- a8 d3 Q. a>> FundamentalUnit(Q5) ;
1 J. r4 ~, @" d& Q/ a; f                  ^
9 u  V7 f0 ]% @& RRuntime error in 'FundamentalUnit': Field must have positive discriminant


>> FundamentalUnit(M);
/ v' m3 j# \) H+ \                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
  j, V6 z2 C; c2 A9 n6 p9 |
3
, `& F9 A- |8 M% k8 j" x# y3 ~
. @3 u3 q" D* |1 j>> Name(M, -3);
       ^
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
$ N3 w9 ^3 c" L1 |
1
2 F5 s! Z- I' G7 @& bAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
( d9 ?" Z0 \5 `1 `; yAbelian Group of order 1
( e2 E  B! ]8 x. YMapping from: Abelian Group of order 1 to Set of ideals of M
1
+ [" N) O7 U! ?. H+ Q4 Q9 @1
) h/ M- z- u- [4 r* u, ~9 YAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
: E4 \3 C9 y4 X( r7 X6 G* Dinverse]
6 m- E  r9 ?! F5 `1
+ w! Z$ \  y3 G7 i) }Abelian Group of order 1
* {, }# C  H& W7 M! t9 [0 V; SMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule [no inverse]
. R% }( z2 \3 v1 \" {/ E* BAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
$ X  s/ s5 f* f5 S2 G. w-3 given by a rule [no inverse]
5 R+ V2 e+ j6 F9 E7 W3 b# \# pfalse
9 g) c# D, ]" G& \4 V1 }% r  ]false

孤寂冷逍遥

lilianjie

% `8 m0 I3 h, p# w, G, ^! Q& UDirichlet character
Dirichlet class number formula
( t( }, ^  W; X, ]- k% [2 v
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
+ `3 q# j& u+ x$ f+ `
-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，
, p7 o7 n: _  `h=-6/（2*3）*Σ[1*1+（2*（-1）]=1
7 h6 P2 _! v7 T7 ]
-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，
  d3 G, e  C- `) P6 ]3 Z


h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2
! j8 s& }4 _& T+ ~
4 N% q# _# _; d" ]& M

-50时  个单位根                          N=200

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

% b4 @5 x7 v% O1 r
F := QuadraticField(NextPrime(5));

! X4 d2 q# @3 M6 |9 iKK := QuadraticField(7);KK;
K:=MaximalOrder(KK);
5 R* ?2 Q1 r, |) ]3 d& Y) ?$ |Conductor(KK);
ClassGroup(KK) ;
* P' l; k" V* u( a9 p. nQuadraticClassGroupTwoPart(KK) ;
, T2 H' A' I  b( t" l9 GNormEquation(F, 7);
A:=K!7;A;
B:=K!14;B;
" X( W: V  x0 |2 o( N1 A# ODiscriminant(KK)
- h; ]( f7 N7 N4 }$ ~
Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
28
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of K
Abelian Group isomorphic to Z/2
+ Q1 ~, R% J4 W4 m+ v* j7 L- r& nDefined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
  y8 X- m; m8 M; O$ i: T% BDefined on 1 generator
+ n" B8 f( c, y2 l  b+ N( zRelations:
    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
* x- e8 ?, ]$ ?/ l8 ninverse]
false
9 r1 D- o) V/ C0 Z, I7
14
. k! N( d8 t0 {8 z  _' G: H28

lilianjie

" A8 H; M) u6 w0 r9 c


5 j* ?$ B2 @% U! R$ ^

; V" N: H; q" F3 U% c$ o

分圆域：
; D: H. I' r0 O2 e. a+ O8 ^* uC:=CyclotomicField(5);C;
: u! Q1 M& T$ u4 VCyclotomicPolynomial(5);
C:=CyclotomicField(6);C;
CyclotomicPolynomial(6);
CC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
MinimalField(CC!7) ;
MinimalField(CC!8) ;
MinimalField(CC!9) ;
4 M0 @" B2 N$ c2 G; nMinimalCyclotomicField(CC!7) ;
: X) u2 D, d0 i5 c1 lRootOfUnity(11);RootOfUnity(111);
Minimise(CC!123);
  d& ^" L" J  N$ LConductor(CC) ;
7 X9 B5 M/ R  U% Z8 VCyclotomicOrder(CC) ;
9 n1 S7 K$ h8 X' R& L
CyclotomicAutomorphismGroup(CC) ;
. g; {& K) ]/ B* R
Cyclotomic Field of order 5 and degree 4
; v4 E- K% ^7 i! X: a( R* w/ |! [& h' N$.1^4 + $.1^3 + $.1^2 + $.1 + 1
5 n4 h. ^$ m) k. w7 qCyclotomic Field of order 6 and degree 2
6 U0 n6 y2 \; x- B+ F; Z$.1^2 - $.1 + 1
Cyclotomic Field of order 7 and degree 6
! _. ^" b8 l0 ?) \9 t6 u6 }$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
2 `& b4 `, g2 U+ |7 Y" aRational Field
  M3 @0 \6 f7 Z" r& i( T- yRational Field
- m4 X6 ]% V+ p1 H0 w( g7 T* K2 R4 ?Rational Field
zeta_11
" `) _' T% V9 I1 C* q/ T) Pzeta_111
123
; c, w5 Z" q$ l# I8 s7
7
. z2 q: T: V2 l4 m- C7 J- s4 sPermutation group acting on a set of cardinality 6
+ C, ~9 N+ q! `/ _2 B" V9 @Order = 6 = 2 * 3
6 c' }; o3 K2 h* P/ m3 s) H7 L    (1, 2)(3, 5)(4, 6)
    (1, 3, 6, 2, 5, 4)
( B# [& U( m$ ~$ DMapping 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: $,
3 S  T! F* {5 c) g3 }Degree 6, Order 2 * 3 and
6 h  @* D% R7 m# P; ?9 _. I7 BMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
/ F  j' C) P0 B# U" V9 r. BCC

lilianjie

* S+ X! n, R0 D; `3 |5 p5 y" g

lilianjie 发表于 2012-1-9 20:44
* b$ E9 F  b' y$ M0 M$ }分圆域：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

分圆域：
分圆域：123
1 P4 L9 I( B5 d8 |
R.<x> = Q[]
F8 = factor(x^8 - 1)
: Q1 |' y: H; ]$ z' g* U: VF8
! ?4 W) Y5 K' v3 U
; y% e- H2 k1 Y& ?5 t6 d: x- F(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

Q<x> := QuadraticField(8);Q;
3 a% J: @9 G7 `C:=CyclotomicField(8);C;
2 K- B' g# z/ ?# t- [" aFF:=CyclotomicPolynomial(8);FF;
, c  e& w5 g% H0 v) ^7 s
; z( }$ w( o$ m4 HF := QuadraticField(8);
" O1 {! z8 l  p% TF;
D:=Factorization(FF) ;D;
% Z( c9 _; L/ y: I3 N* DQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
0 H$ \& Y+ F6 S2 d7 z6 l) c$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
    <$.1^4 + 1, 1>
+ r# Q$ e. Y$ t7 T9 ^" Q3 x]
% M1 d! V3 j1 p4 _9 {8 P
1 H% O; K8 \8 BR.<x> = QQ[]
7 [$ M) l! o7 G: j) ?F6 = factor(x^6 - 1)
# c- A1 g2 }7 J; M" Y! yF6
' ~9 R1 r  i3 U; O7 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)
- F7 N, Q3 }- C& o2 p1 G& V
$ V: ~% E* V+ i9 `/ p( I1 k+ I$ zQ<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
* n; y' l$ U9 G% i" ?0 iFF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
/ N& l( q  M2 {7 j6 x. g) X3 sF;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
9 {. E9 @  c) h4 ~% F  R0 nCyclotomic Field of order 6 and degree 2
# ]( H# f2 c% W: j/ O/ v0 l4 J$.1^2 - $.1 + 1
, h7 h  R! H! V( i" d+ k- O0 qQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
# d" B- Q5 \* M4 X7 ?6 A    <$.1^2 - $.1 + 1, 1>
]

0 k! q) d: |5 _R.<x> = QQ[]
F5 = factor(x^10 - 1)
  r$ C) y4 v# z- H' PF5
, C6 h1 J0 W7 ^9 L6 C  A- N! Q(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
; z, ]0 R& }# p5 p4 u8 s/ F3 e1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
" E) ~* {( ^2 x9 R0 B
Q<x> := QuadraticField(10);Q;
) i4 {$ k* s" o, y9 l2 @5 DC:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;
  z/ q9 G+ N- s5 v
F := QuadraticField(10);
2 ?  X% M+ d$ N  ^* q2 |9 KF;
, p- e# [; i( v# lD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
4 c5 k) C& D# X5 T) Q  D0 i4 ]$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

lilianjie1

分圆域：123