虚二次域例两（-5/50）

lilianjie

2 `% ~8 Z# ?/ O* t6 z9 Y# C9 K* k
  }; i# _; {7 k  gQ5:=QuadraticField(-5) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
% D  o6 l, C" @( oM:=MaximalOrder(Q5) ;
M;
( z1 B& z' Q' m# U' jNumberField(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;
8 f# `) [! {; `1 d" HIsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
! g, j0 ]: L) r6 o+ kIsQuadratic(S25);
IsQuadratic(S625888888);
/ s. ?/ y! V9 I# I, P6 z- @Factorization(w^2+5);  
Discriminant(Q5) ;
) g# E2 ]5 f  I2 e, I# F' Z. F% T" ~FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
0 i! c. b+ D& l
/ [1 X  M2 D1 p6 r( ]( l* D8 }Name(M, -5);
Conductor(M);
ClassGroup(Q5) ;
3 _/ R) b6 ]0 p* BClassGroup(M);
/ g6 G( p) t( _ClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
  N3 ~$ L! G+ W2 c4 d3 x# e
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -5) ;
# V7 x8 \6 r. }NormEquation(M, -5) ;
) c  A- L: B8 \! b& R; g* WQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
; |1 \+ Z. M( R% Q& gEquation Order of conductor 1 in Q5
: H" O/ H! h& d: {+ zMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Order of conductor 625888888 in Q5
* t5 j/ ~+ v! O5 ~6 q; Qtrue Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
true Maximal Equation Order of Q5
. r( c" Y6 _) o% d9 Etrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
- m4 o& C1 \' ?6 g; g2 `true Order of conductor 1 in Q5
[
2 z6 [0 [3 e' s. A2 \2 v8 A    <w - Q5.1, 1>,
1 C9 t. ^+ P$ }4 {8 o    <w + Q5.1, 1>
]
-20
" @$ f+ h/ m8 N7 I# o4 O9 ~6 j
0 @/ w* p# c; ]& c>> FundamentalUnit(Q5) ;
5 N. l6 `% K0 I1 L                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
, z  [. I3 S* K
0 H/ u2 E% ], m2 |
1 |7 K4 [' R1 _% T& _9 I>> FundamentalUnit(M);
  Q( v. L% V* R' K) o                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
. _0 \: O2 N) v, ?
( Z9 i7 p7 m; U8 W20
2 i1 y8 i/ S5 H3 V
>> Name(M, -5);
" L( e5 H3 |. E8 F6 I1 ?       ^
Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]

+ x  p+ u- v: I5 a1
0 H; c! i/ _  N+ B$ R$ F0 ^Abelian Group isomorphic to Z/2
Defined on 1 generator
4 e3 l/ L5 A: @, r( MRelations:
  x9 _, u0 l, C    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
6 g2 H" n7 K; p$ H. j    2*$.1 = 0 to Set of ideals of M
Abelian Group isomorphic to Z/2
Defined on 1 generator
+ z+ L+ Q& C( uRelations:
4 W+ A  M8 i# Z- o0 R% p    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
4 x# ]% k" F" M' W4 JDefined on 1 generator
5 V! ]6 s! I0 d- X4 \2 M$ f6 URelations:
    2*$.1 = 0 to Set of ideals of M
3 H7 H9 b. `! x) L- G) V2
2
Abelian Group isomorphic to Z/2
Defined on 1 generator
% }/ P$ J  X/ ORelations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
+ W  y) a* n' O- w8 h0 q* D( HDefined on 1 generator
! p* ~3 P$ U: ORelations:
    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
; _/ D: R, [: [7 n3 c! e# D2
1 L! E, C3 u! ]: jAbelian Group isomorphic to Z/2
4 v% C+ ]) B% X$ t2 t0 jDefined on 1 generator
, e2 t+ j5 O' Z- M# CRelations:
    2*$.1 = 0
0 O& u, r2 U+ A0 JMapping from: Abelian Group isomorphic to Z/2
1 P, @6 ?3 m. S' iDefined on 1 generator
9 }7 {! r3 u4 {8 V$ dRelations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
Abelian Group isomorphic to Z/2
9 `' ^3 N" ?1 r" ~3 GDefined on 1 generator
/ I% S: e/ W  a$ j/ o) Y4 vRelations:
! b6 J5 ]* e2 l  z0 V4 S    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
/ B2 P  `* m9 Q& T7 x+ V+ T0 IDefined on 1 generator
Relations:
; }+ ^  x: a; M    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
% y$ `/ U3 }* y7 `( _inverse]
false
false
==============

; e8 a9 G' r- V9 g6 X5 P+ F
Q5:=QuadraticField(-50) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
# o) c6 J6 E6 p2 VEquationOrder(Q5);
. x4 a! d/ O+ \/ Q* R  k/ |M:=MaximalOrder(Q5) ;
6 @7 q" W' S* }6 Y4 ]M;
( o  K+ R# M  S# S& @NumberField(M);
  e. l$ y, {1 ^, o' d% aS1:=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 c4 c( C5 y1 |- J4 z: F* U9 BIsQuadratic(Q5);
0 u  t* V3 i! s& l; q+ u) dIsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+50);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
" t2 o, A8 |0 m: g' Z( K( TFundamentalUnit(M);
, H4 G  E, E8 {- Q# hConductor(Q5) ;
6 a* _% |. l6 c9 |9 `
# h) ]" ~2 p+ a% Z1 R- R- Z2 JName(M, -50);
5 C+ \- ]* w: Z( l6 k" H( qConductor(M);
ClassGroup(Q5) ;
5 ^' h1 C* t6 Y2 k1 l7 V2 QClassGroup(M);
ClassNumber(Q5) ;
0 Y% `+ t# |+ Z2 AClassNumber(M) ;
& x1 U% x$ z( w/ }9 RPicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
  }% r) U* l' p% iNormEquation(Q5, -50) ;
! B% {' {% \! Q" {6 ~; {, G  n$ p- gNormEquation(M, -50) ;
1 T' v4 `; L8 L, D- i
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
  F" O/ r& T9 O4 ]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
+ L9 I2 \3 g. pOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 2 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
[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
8 d, T0 |; x' W7 P0 w]
. J  F; Z7 p2 O8 N& _+ M- x  [-8

>> FundamentalUnit(Q5) ;
4 \+ N6 C$ d: O3 [) a6 j7 W                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

- x: ^2 N( N' K9 l# w- Q
>> FundamentalUnit(M);
+ Q* Y' O5 V+ B: ?/ |2 s' j                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

8 F1 J& P  G" o. b8
* O& |! u2 u) G* |
>> Name(M, -50);
       ^
Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]

9 {- g1 j% T5 b8 F) i% U( `  J1 C1
Abelian Group of order 1
& |; ~' I- D4 l" TMapping from: Abelian Group of order 1 to Set of ideals of M
9 [/ u$ W( k# MAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
4 k$ E- S9 D& }5 Z1 u' G1
1
8 @$ F/ h# Y$ T+ o) ]- N% N9 ]Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
5 ~' V6 ?! J. s$ Y" d$ C; w* Sinverse]
) f: @; v, g% M! U& v4 L9 p1
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]
0 i  \+ D/ g  lAbelian Group of order 1
; a3 E! }  c! A( @4 ^& M: ~Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
/ {2 ]: A2 f7 [-8 given by a rule [no inverse]
false
6 O6 v& a( I8 H4 f/ M. Z) [false
+ F! g' A" M6 F* z0 k# D+ {

lilianjie

看看-1.-3的两种：

# q( N$ j; H4 ^, y8 H& mQ5:=QuadraticField(-1) ;
9 F. j% N2 x: K+ n4 _Q5;
- Z1 w" \: {! N" \$ [
) @) F9 u+ c5 Y+ O4 @1 e; `6 TQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
: S! O: ?9 \0 F* sM:=MaximalOrder(Q5) ;
M;
; m; I8 L% I5 q. u( i9 [! G+ zNumberField(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;
) ^, M$ c7 j3 WIsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
$ b5 ?8 W# Q$ H" lIsQuadratic(S625888888);
Factorization(w^2+1);  
7 h! w0 l7 o% C& R" d0 o/ E$ kDiscriminant(Q5) ;
2 `7 c: \! X  b" T& ~FundamentalUnit(Q5) ;
  p! }) e! |+ X1 x9 c6 k! ZFundamentalUnit(M);
) O* X0 ?8 `' f1 n4 J' C/ E" k& A) [Conductor(Q5) ;

  C* }( u1 _0 x. r7 {$ R" qName(M, -1);
Conductor(M);
' Y% v4 a; i/ u: \3 ^. Y6 p$ H7 TClassGroup(Q5) ;
ClassGroup(M);
- T1 w$ c$ z0 D3 R2 G% x: ?  |ClassNumber(Q5) ;
1 G2 ?& a) y- P; ]) MClassNumber(M) ;
PicardGroup(M) ;
, V1 R9 O  ~' t# M( NPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
' C) v% C8 _$ k& cQuadraticClassGroupTwoPart(M);
NormEquation(Q5, -1) ;
! h9 x, B- ^% D! mNormEquation(M, -1) ;

Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Univariate Polynomial Ring in w over Q5
: P! B7 p; e5 G: l! uEquation Order of conductor 1 in Q5
9 R& o5 R3 c; o$ o! Y' B  S- d- ]6 w1 fMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
' C3 E; y' D/ c4 ?( R0 zOrder of conductor 625888888 in Q5
. I6 s$ e! `* i, {true Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
5 \# I* e: m! T( t) ]$ Qtrue Maximal Equation Order of Q5
8 c4 X8 s- M& f1 x3 `& y6 G# U& Q+ {true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
+ d% ?3 V: ^# ][
5 T8 G9 s% o7 C) Q, I    <w - Q5.1, 1>,
0 _9 M6 f0 ~4 V1 ]$ G" P    <w + Q5.1, 1>
* J/ B9 B( y8 n* i]
' N, V) g+ {2 j2 Z+ T7 U  L) u-4

7 [/ b' q' y- F# M; ]>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant


>> FundamentalUnit(M);
3 \* C/ V' d! d& U4 d                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

# H) M% ]9 y7 O$ V7 h1 P4 C& b4

>> Name(M, -1);
5 X7 r% \6 v+ _0 K' F1 W2 m! Z4 v* ]. z       ^
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]
7 Y! Q/ D! l& b" o' a
1
Abelian Group of order 1
# f- ]1 b9 M, W" aMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
$ ^6 Y8 }' Y" Z6 k1 X- PMapping from: Abelian Group of order 1 to Set of ideals of M
: I4 \4 h7 P+ _' i! T+ b1
7 g3 l$ c) M$ g& e! G; I7 J; I5 l1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
+ }# b" p6 h2 k( R5 y: ~/ Sinverse]
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 given by a rule [no inverse]
$ q0 r' A) @' xAbelian Group of order 1
: \6 z4 Z8 n0 f" EMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
6 e9 K5 C' e  w$ K-4 given by a rule [no inverse]
false
false
===============

Q5:=QuadraticField(-3) ;
: o# I. j4 X% {' C* |6 MQ5;

Q<w> :=PolynomialRing(Q5);Q;
+ y$ I; y$ N; c9 W% N$ ~EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
( c. G4 H4 P' v* U" w7 ]: D1 hM;
& u( c1 j. @8 T. [8 Z8 G) G7 ANumberField(M);
+ ?1 W1 Q, T: i: `; n( FS1:=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;
% y9 [9 }$ L2 M6 ~* r: F1 w- e4 `IsQuadratic(Q5);
' q( r5 |9 ^0 J2 \IsQuadratic(S1);
IsQuadratic(S4);
) z2 s$ k8 ~& GIsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+3);  
; E# X! d: }8 T' T7 g3 HDiscriminant(Q5) ;
/ l' k( F7 C. h# N. B2 m5 oFundamentalUnit(Q5) ;
8 \6 r5 b0 B  q: R" [5 ?FundamentalUnit(M);
Conductor(Q5) ;

( R# }4 q1 R) ]7 IName(M, -3);
/ o8 ]! V5 K4 j7 }) m% C4 uConductor(M);
" d. n0 g8 h# d) AClassGroup(Q5) ;
2 v1 u" u; i% D& Y  vClassGroup(M);
* \  [, [* ~$ g* w; dClassNumber(Q5) ;
; ^0 Z6 o  y# O* nClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
# \* M' ]3 r* T& f1 s
3 Y+ m4 s; H2 `& q1 t9 J9 tQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
. ]/ O* [5 o) x& g1 s  B- p/ ~3 YNormEquation(Q5, -3) ;
NormEquation(M, -3) ;
8 R/ k+ Y" T: O* s
; J  B( Y& {5 J. hQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
' ]; f* d# u/ w# `2 JUnivariate Polynomial Ring in w over Q5
2 Z) n  \$ g8 j! Y, v) y$ kEquation Order of conductor 2 in Q5
4 |/ P+ S4 P) N2 r' XMaximal Order of Q5
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
/ \  p, W- E8 @2 q+ I( S% Z. KOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
7 q8 y! |0 R: ]5 ?) ?! \true Maximal Order of Q5
true Order of conductor 16 in Q5
true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
4 W$ c) n3 t+ v( Y[
    <w - Q5.1, 1>,
$ |! I3 z6 f: ?8 r    <w + Q5.1, 1>
* |2 Q5 F+ K* K) P0 P5 n6 P+ L: Z]
; O* X& x0 I1 m1 f-3

>> FundamentalUnit(Q5) ;
; K2 V6 [6 x' c* W9 z5 |                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant


>> FundamentalUnit(M);
9 r) x7 n/ ]% Z+ d  X; x9 a6 j                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
8 V& y4 |' U9 m+ C3 U
, g3 U3 w$ D! Z3
# G6 }8 X$ F0 a* K
>> Name(M, -3);
- c5 I$ Z9 Q* c# G       ^
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
; n& w" V* [8 v: {* S7 M0 h5 t% u
! M( X  `0 @1 J5 g1
Abelian Group of order 1
. n. p) e) z$ }* |( }4 \Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
' g3 b7 ]2 j& O  _8 z$ o1 xMapping from: Abelian Group of order 1 to Set of ideals of M
1
1
Abelian Group of order 1
/ L& U& x. M3 F; kMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
- c4 Z( v8 {- [. ^( I8 k& A* i4 C1
' C, B0 R8 z# j& {8 HAbelian Group of order 1
# j7 ]$ ]# ^7 fMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
, F) f! t( Z- f# L3 ^-3 given by a rule [no inverse]
Abelian Group of order 1
. V9 C; R! ^  R- I. oMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
) s% D$ \2 j  Z8 g* _5 ]8 D/ _-3 given by a rule [no inverse]
false
false

孤寂冷逍遥

lilianjie

Dirichlet character
% S" `0 Z9 U3 Q5 ADirichlet class number formula

0 i6 p9 o' f" B6 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
5 m/ |( }4 C) L" F4 B' e
# P3 G# |" o$ X6 S' u5 O$ ?-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

6 |, }9 g. m0 t: Q0 P-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
: {" H2 ^4 ]( Uh=-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，

0 i' S6 a/ N" |, x$ `
/ V; V  ^& G" d( l" n1 ]! o
  d" F+ \$ f8 i3 \. B0 hh=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2
0 J# D* j3 Z/ i3 u

9 o  G/ v& P+ ]+ `, V: @
1 c$ [8 p! x- d  c! f: r-50时  个单位根                          N=200
1 y8 J2 j! Y( R3 t- n) y$ o

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

& `/ I1 |  @" G; C4 W  x1 o
F := QuadraticField(NextPrime(5));
1 j7 ?3 d8 i* ]$ p; n" n
( }" R: l, ^' k. L# b* O  L1 [KK := QuadraticField(7);KK;
K:=MaximalOrder(KK);
. c( U" ]7 ^3 VConductor(KK);
; \; U" @, Q% [! IClassGroup(KK) ;
7 ?0 N8 N/ o5 y  \7 Y  q1 GQuadraticClassGroupTwoPart(KK) ;
  j. o7 H% X+ e5 _; tNormEquation(F, 7);
A:=K!7;A;
B:=K!14;B;
3 L7 v- J6 P  o: G9 TDiscriminant(KK)
# m. }4 I7 X$ a
* e; L! @, R0 Y9 e, RQuadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
28
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of K
% n  i, e/ n& O: P5 u+ A, MAbelian Group isomorphic to Z/2
% Y% V0 M  `# S$ @) O- P4 {Defined on 1 generator
Relations:
/ I/ o) j$ P) @, u; c7 r* i$ G/ `. v    2*$.1 = 0
+ w7 X' v+ k0 F( b# J! XMapping from: Abelian Group isomorphic to Z/2
) W! w7 g/ I& K3 H+ t1 w5 uDefined on 1 generator
Relations:
) B  N1 T# t& s5 W( f! F. z  @( X    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
& p* m/ v+ y; [4 @* u5 finverse]
: j: r+ Y, ^  T3 Yfalse
7
1 e( ]# K7 S0 {) c& B14
28

lilianjie

4 Q9 s9 p# H; r

( a& f& ~9 d! K: K
9 y5 j! v$ q! ?- P: y
4 {4 }5 t6 I1 H( A- a/ h2 @分圆域：
7 C* h' v7 L! Q* G* CC:=CyclotomicField(5);C;
' B# M( _6 R# T6 ~0 M1 @4 eCyclotomicPolynomial(5);
C:=CyclotomicField(6);C;
( m  j) e2 _( W4 pCyclotomicPolynomial(6);
0 ?0 U$ b2 R. p5 d( z5 o2 BCC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
+ U5 M6 r3 M) _- hMinimalField(CC!7) ;
MinimalField(CC!8) ;
MinimalField(CC!9) ;
8 `5 P" [2 K  L% R9 T  AMinimalCyclotomicField(CC!7) ;
RootOfUnity(11);RootOfUnity(111);
Minimise(CC!123);
Conductor(CC) ;
: b0 E9 [6 P8 n8 PCyclotomicOrder(CC) ;

6 m2 v. a0 x+ V* HCyclotomicAutomorphismGroup(CC) ;
- @: w, X+ z" g2 J4 D7 {# x6 `. H
Cyclotomic Field of order 5 and degree 4
$.1^4 + $.1^3 + $.1^2 + $.1 + 1
Cyclotomic Field of order 6 and degree 2
$ ^4 Y/ _+ T2 [$.1^2 - $.1 + 1
Cyclotomic Field of order 7 and degree 6
+ m2 u# k, l$ b, H/ o$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
Rational Field
Rational Field
$ M" P; R, ?$ l. E3 LRational Field
( _" p/ W. {3 yzeta_11
+ ~$ `4 u  Q1 a  czeta_111
; e  j: I4 O: ~$ T; U123
7
7
4 J1 p% X0 z; N0 y4 r, k: ^1 ePermutation group acting on a set of cardinality 6
/ n, [5 a1 q5 K& O5 F5 _Order = 6 = 2 * 3
( M5 B! s/ c0 M- r2 z( i4 r9 a    (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
Composition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
$ D2 i+ y1 s" c" E  XDegree 6, Order 2 * 3 and
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
/ |  E* ]" \( e1 ^CC

lilianjie

: H. e4 X1 A" Q9 _% T% g9 m( O

lilianjie 发表于 2012-1-9 20:44
分圆域：
C:=CyclotomicField(5);C;
. P; C0 p, I9 eCyclotomicPolynomial(5);

  R% W& a8 o  L6 n* ]4 J
分圆域：
5 ^# x) {8 [. C3 I- B! d/ a* P分圆域：123

' R: L: u2 h5 q# p# W: u+ q% v& NR.<x> = Q[]
4 A" w1 h9 A8 L% L" DF8 = factor(x^8 - 1)
F8
) z& u3 W# Y. q# k
8 A! i. y1 S: G- l9 F8 H(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

& P6 t; B; h0 ?' _+ c- B" QQ<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;
! W$ K6 J: ^/ @2 y2 b* r% Y! p* Q, [+ H
3 ^% Y- F9 w  @F := QuadraticField(8);
& Q: N8 X7 e" w: x6 t+ CF;
8 H5 ?8 u* F3 T+ WD:=Factorization(FF) ;D;
  F; k# a+ Y5 @) z9 O2 Z5 _- SQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
$.1^4 + 1
* _1 f  f: C2 r7 ]Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
" s3 r, c8 H! F6 l3 [[
0 ~1 {$ S2 W! ]1 l    <$.1^4 + 1, 1>
]

3 S& S$ D# ]9 a- ]( p9 c/ \3 JR.<x> = QQ[]
F6 = factor(x^6 - 1)
( D- d$ c' f2 K8 PF6

(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
$ H  @# b: q* Y- g7 T& h! r  `' k
, X7 H; Q$ C/ |+ W$ V1 J8 e. UQ<x> := QuadraticField(6);Q;
+ Y3 U# q6 k4 q( g" a7 [C:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;

* M9 Z' }# U  ~0 d+ r6 o! Q' l. TF := QuadraticField(6);
& h: ]1 K# P/ \& |; o# a, mF;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
9 }( ?$ F7 s7 C) Y) WCyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
  R, f! i% K2 I( o: f3 @1 A0 m[
; u- }0 g( l; S) I( i    <$.1^2 - $.1 + 1, 1>
, C$ _) G3 z6 a* x% U! Y& f, T]

R.<x> = QQ[]
F5 = factor(x^10 - 1)
F5
- o6 J* H& o" d! ?7 }: _4 j% t(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
/ x: |$ c2 `$ p; u. b6 {2 {  ]1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
8 s# e4 m% ^/ i0 a2 c6 d7 M
Q<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;
1 t' O" w; G& Q( n. A* ~
F := QuadraticField(10);
2 r/ y1 z; h( R' QF;
D:=Factorization(FF) ;D;
" I/ `+ e1 f) w+ RQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
) A/ [' x0 b, x3 |! V  U/ n$.1^4 - $.1^3 + $.1^2 - $.1 + 1
9 n) S/ ]3 O  Q$ J% nQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
# Q2 v$ `7 B% e- q3 P    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

lilianjie1

分圆域：123