虚二次域例两（-5/50）

lilianjie

7 T2 ^; n4 e1 S3 e
Q5:=QuadraticField(-5) ;
& I% o& a, M' \) {Q5;
! {) i; V* `: B4 v- v' T7 a8 \* V
- y! s8 v: e' K( p/ FQ<w> :=PolynomialRing(Q5);Q;
" d4 E  Z! F2 _, N/ a& K' _EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
) u* t2 h: c" i; ~* U' xM;
NumberField(M);
& l: J# e7 l- }7 R* z9 j6 O1 zS1:=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;
$ m7 ?2 H. N: RIsQuadratic(Q5);
IsQuadratic(S1);
% [5 s1 n3 F4 C1 c$ W+ JIsQuadratic(S4);
# _- D0 w1 p: v& B6 k8 KIsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+5);  
* p: u% P6 P. q2 ZDiscriminant(Q5) ;
# ]7 z3 M# j6 x& FFundamentalUnit(Q5) ;
+ _7 k8 U/ Y% S: o/ @( n3 ^FundamentalUnit(M);
Conductor(Q5) ;
7 ~8 v9 v8 ~! [0 B' T
/ U; T- ]# m& U! }1 BName(M, -5);
( l- L, k! J. V" W9 L( J# bConductor(M);
4 ~; C0 S- j7 V: w3 I1 SClassGroup(Q5) ;
' d3 W1 E; s' [: m% Z2 F7 t% ?ClassGroup(M);
ClassNumber(Q5) ;
( x7 H/ f! y8 U2 L& [) KClassNumber(M) ;
PicardGroup(M) ;
# K; J! t1 `( S; P) r6 QPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -5) ;
% }* K4 y6 K, H% p# LNormEquation(M, -5) ;
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
3 p8 C% S3 F4 k; [+ Y: ^7 {, `Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
6 l3 \" v+ P2 y% cQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
- |. p# ]; \& r8 Y8 S+ fOrder of conductor 625888888 in Q5
* p2 U3 I. z/ p" ^9 n6 D, Mtrue Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
. I; Z2 z" w' [6 R+ Jtrue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
2 [- o, m  l# V1 u2 g7 Atrue Order of conductor 1 in Q5
5 {% i& R5 E& c7 Xtrue Order of conductor 1 in Q5
; V/ O& [0 M6 ]+ O3 |3 ^/ F[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
7 U2 N4 }* I  X7 g! A0 D% j]
-20

>> FundamentalUnit(Q5) ;
' ?, K) k' q6 S3 W                  ^
$ ^3 d3 B4 ]8 L6 O! E  i8 w, O% WRuntime error in 'FundamentalUnit': Field must have positive discriminant

: u1 v, J; y& ]
>> FundamentalUnit(M);
; G; Z7 m, |2 h: K' ~                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

20
  z$ H; J( o# H/ {5 z
>> Name(M, -5);
       ^
Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]

+ z" N) W! F4 D# }; K' I6 F6 P1
Abelian Group isomorphic to Z/2
/ m1 P( m8 u3 S+ ?Defined on 1 generator
Relations:
9 V0 h5 i0 Y+ N2 z    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
& w& _& B  c, x! \- I  J/ _Defined on 1 generator
2 r5 u1 ^9 c4 F8 k0 z. o2 q0 sRelations:
4 `& a. H$ L. r( M5 s    2*$.1 = 0 to Set of ideals of M
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
) A) T3 G) N7 N+ S5 `Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
4 X: H6 K! t$ `& q/ \0 kRelations:
    2*$.1 = 0 to Set of ideals of M
2
2
9 i2 n6 O1 x8 K1 C6 m1 G' w2 r! b9 j  `; k* OAbelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
$ v# [4 L5 r. E; c5 L4 N% JMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
* V0 r& f2 V% l6 Z, G& z% n8 D: e2
Abelian Group isomorphic to Z/2
( M. ?: u8 G2 M4 J9 u. E% q8 q$ PDefined on 1 generator
- ^# X$ j. E/ I, gRelations:
    2*$.1 = 0
% Z/ ^* Y6 p5 O" e4 O3 GMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
, z3 o9 T6 v' t* p( Hinverse]
# ?5 A& D5 K$ o# e  E4 H/ xAbelian Group isomorphic to Z/2
" ~5 D: L. [1 p7 r+ |% dDefined on 1 generator
Relations:
9 C9 R- Q. c4 z* r( k& A6 x    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]
" E4 `: _0 j% t" s. j0 n; o% Sfalse
2 O) }* n% O7 V4 D+ K% bfalse
. N4 l" W) r) c+ i==============

5 Z8 |+ C# f& X
- y  e& m* \- t0 Z* {Q5:=QuadraticField(-50) ;
Q5;
! |% I& Q3 Z, V, A0 P
Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
- ^2 E: `& j: u6 O2 ?2 p" a0 iM:=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;
9 u& \$ _! Z3 C: |+ z- B7 tIsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
- }& M0 c! }. `4 D3 eIsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+50);  
3 N# M  }  H2 }# ?. Y5 L* d5 cDiscriminant(Q5) ;
% u+ _' C! l6 k4 `  bFundamentalUnit(Q5) ;
FundamentalUnit(M);
! Z" x+ ^/ ^  y, F. X* ~6 ^5 l+ mConductor(Q5) ;

Name(M, -50);
Conductor(M);
ClassGroup(Q5) ;
$ `7 ~7 m0 ^) X& S0 C7 [ClassGroup(M);
ClassNumber(Q5) ;
4 Z1 k/ B1 x1 }$ P6 U* Z/ RClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

( @( A% w9 _/ wQuadraticClassGroupTwoPart(Q5);
1 y4 _1 X. e+ m' o" `QuadraticClassGroupTwoPart(M);
7 F6 ~9 V% o- V8 [' J6 XNormEquation(Q5, -50) ;
NormEquation(M, -50) ;
) A" i6 K% A" z8 b, [
) H: _5 q; T; tQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
8 |1 R0 E7 M' F6 w" ZEquation Order of conductor 1 in Q5
: n% l0 O" F$ B8 DMaximal 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 $.1^2 + 2 over the Rational Field
true Maximal Equation Order of Q5
# Z' K7 n$ e1 A/ |0 \/ a3 x& btrue Order of conductor 1 in Q5
/ S) t& t; d# g" ?true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
! x6 F9 L9 O7 v" O( ?+ b    <w - 5*Q5.1, 1>,
) ^- a  V' t9 ]# q% z. u    <w + 5*Q5.1, 1>
4 n$ H* U6 r8 R1 E4 }" c5 G]
-8
$ p" t! ]. o, v3 t* ^, [2 ?6 S
1 E! O! U8 x$ T) i! b6 X3 K& y>> FundamentalUnit(Q5) ;
4 \3 Z( u0 P7 z3 o  B: b+ ^+ k                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
: ]( y, n* B$ D* v

. h6 V) f+ |' N0 T>> FundamentalUnit(M);
# v0 G7 _( h: q2 X) p; A                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

7 P/ o" w! c" Q3 _8

>> Name(M, -50);
) L2 W5 t4 _0 m9 \; G- F6 p1 b       ^
Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]
# ]+ i: X! X, y
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
; U1 T! g2 L. _4 m: U9 t, T4 D; mAbelian Group of order 1
! E/ _, {. s& q+ ]Mapping from: Abelian Group of order 1 to Set of ideals of M
. h' l4 ?, d8 D4 A  Q5 p1
+ C' Z0 z! ]+ Z& w3 Y& P8 ^" G1
4 c% b6 E! F3 P2 D; zAbelian Group of order 1
; q) S) O/ r; v7 ~+ PMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
* I2 `  ]& v2 m$ Ainverse]
+ `0 t0 h& e, a3 y, W, e; e1
Abelian Group of order 1
0 p( M0 c4 B: TMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
% n" K' m2 I/ w3 [$ h-8 given by a rule [no inverse]
Abelian Group of order 1
7 ]( m# q5 o& A& OMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
" F; L8 f* _* E! w' L-8 given by a rule [no inverse]
false
false

lilianjie

看看-1.-3的两种：
) w6 {/ M, M) b2 n0 F/ r
Q5:=QuadraticField(-1) ;
1 r- f6 ]* ~/ J6 Q* b- dQ5;

1 y% [- r- n% e" mQ<w> :=PolynomialRing(Q5);Q;
' h3 {. P- G/ aEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
0 m8 I' v7 G$ K5 {& Y/ P! YM;
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;
3 A! ~) J: y5 N# t8 ~, z2 bIsQuadratic(Q5);
# i( J' K' A/ e6 P0 oIsQuadratic(S1);
IsQuadratic(S4);
* d6 _: J, d5 [" _+ F1 y5 ]' nIsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+1);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
: e* w1 K/ |' ^0 _9 K0 \Conductor(Q5) ;
+ f3 @; X! t: c0 U8 L' {# J; a
2 m: S) s" m5 f  ]Name(M, -1);
% w2 x& l9 D. B" z" S) ~, ?Conductor(M);
/ m6 _- B+ C7 q) H% @ClassGroup(Q5) ;
5 U4 R! ^% z" S7 e  p5 L& GClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
2 ?; y6 N8 _0 r8 A4 k( A: vPicardGroup(M) ;
! ~3 S0 M9 y* d( z+ ~: LPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
9 t& A. V; r: p9 d( \/ CNormEquation(Q5, -1) ;
NormEquation(M, -1) ;

- W* d, D/ Z: OQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Univariate Polynomial Ring in w over Q5
" K( c  c6 Q& e$ f0 c) f9 v( QEquation Order of conductor 1 in Q5
* v' _  k4 \# s8 @% |# i$ f% ~Maximal Equation Order of Q5
2 _. U$ R, j: }- tQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Order of conductor 625888888 in Q5
+ N' i! I+ H* Utrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
6 P, V! y) p# {+ S$ Ltrue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
: V. q5 m: K/ C) \! b( \true Order of conductor 1 in Q5
[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
-4

' P) U1 f- C& T1 R( e>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
" T& }4 g4 w- s1 U( {

% ?0 m0 \1 p5 c6 c, d5 {>> FundamentalUnit(M);
( c) q. V- K, r. y: w1 J                  ^
& S, E) X3 |8 H, M. ]# FRuntime error in 'FundamentalUnit': Field must have positive discriminant

4
5 O2 v3 r" T5 Y% N
>> Name(M, -1);
% r3 i8 j9 B$ q) Y$ p( q, g       ^
8 T/ H7 ?7 z1 h  |& B7 m# nRuntime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]

& w$ V) n0 [2 N2 B) u1
Abelian Group of order 1
6 q, S  ~/ _+ S1 R% J2 m, eMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
7 N- }: M5 {5 N0 NMapping from: Abelian Group of order 1 to Set of ideals of M
1
+ e) N5 Q% Z2 _4 M9 u1 ^1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
2 J/ V2 g+ B  B" B! D  finverse]
1
  U( \) @: B7 v# \( \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]
4 Q1 d' }" B3 aAbelian Group of order 1
+ w; d' V% L* s: H& Y5 ]0 O7 K# oMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 given by a rule [no inverse]
- ?; e$ ]; e1 I  Z# Y+ nfalse
- {2 f) Y. q4 {9 a7 t, Mfalse
0 x0 x' V* ?) {+ w+ P" Q3 n===============
* w# h0 ?/ N% C/ t2 L
  j/ X$ G* a  Y. jQ5:=QuadraticField(-3) ;
& T* t" p2 U6 t2 Z; z: zQ5;

Q<w> :=PolynomialRing(Q5);Q;
* N( \& Y0 b- S7 ?EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
NumberField(M);
5 L$ x5 l" v3 i. N! zS1:=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);
0 b# q" T6 |# N8 V8 }% L. ^- w$ cIsQuadratic(S1);
' ]+ N) z/ V  jIsQuadratic(S4);
( _) N' ?% D0 _) v4 fIsQuadratic(S25);
4 `" k6 [$ m, ]IsQuadratic(S625888888);
Factorization(w^2+3);  
$ G/ U' i) {4 X, j2 c& m, v) VDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
- }6 I! K- R# n& U2 n3 |Conductor(Q5) ;
# Y, p+ L2 r% W4 i) e
Name(M, -3);
  V1 T& ], d+ B% DConductor(M);
ClassGroup(Q5) ;
0 N. v, f, E6 @  GClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
) U& F+ w# _' Q- `6 e6 Z& \+ }9 a0 C" LPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
9 `/ i: a( n+ M0 k, l! F6 T/ k1 zQuadraticClassGroupTwoPart(M);
2 K6 C/ u  A  J" ]  l/ pNormEquation(Q5, -3) ;
1 ]- y/ X& S& C6 eNormEquation(M, -3) ;

/ j6 b+ ]" J/ k4 AQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
* u" C) S/ q, KUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
! r. U& a! _! ]0 C0 S$ O; LMaximal Order of Q5
7 Y* S  E3 F- i+ j' ]7 WQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Order of conductor 625888888 in Q5
7 B( |6 v0 P- _# \9 G, x5 Htrue Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
* n3 V3 O/ g; N3 u1 ~true Maximal Order of Q5
true Order of conductor 16 in Q5
true Order of conductor 625 in Q5
' g) R, I$ _6 Ptrue Order of conductor 391736900121876544 in Q5
! A: @( @6 }& `[
5 e; h0 m& s! ^% F% a7 y    <w - Q5.1, 1>,
    <w + Q5.1, 1>
' W! ^1 {; Z2 t. X: U]
-3
9 Y* F# J3 d) ?# O9 Z
0 m" f9 ?; ]' J" k  {>> FundamentalUnit(Q5) ;
# {9 M& y3 V  Z) b! s  r                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant


) \  \$ M8 o, Y% a7 {" ~>> FundamentalUnit(M);
; o* F5 q) k/ |2 T$ z/ y                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

2 Q8 w/ C; M3 t6 p: l3

& W# \& R* R/ o' R( b, ?>> Name(M, -3);
3 c/ G& O9 P" m( ]5 n! [       ^
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
2 E# o; l  a+ e9 T0 u) e" c
- A8 {7 t, T7 S/ l) h1 d7 j1
9 i/ t5 [. C8 i% RAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
: N: j& I2 z8 C. w, wAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
; ]2 u) t3 P3 |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
' U# X4 n! L# n# G7 finverse]
1
2 K; \& ~, Z% fAbelian Group of order 1
$ ^0 B  _# m& m/ R5 j) y: ^! oMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
  B9 I% Q2 L& \, Y8 K! ]8 c-3 given by a rule [no inverse]
7 I$ g; L+ Q0 ?6 C( XAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
* Q6 b  {- L% a# v-3 given by a rule [no inverse]
  q5 J/ R' w# N' w0 `false
4 R; r! \; |! f- ffalse

孤寂冷逍遥

lilianjie

0 s+ P* b6 O  y1 ~3 n6 ADirichlet character
Dirichlet class number formula
1 c9 [, v( ~# Z$ e, T8 l: U
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
) W4 s4 ]; S* i" i; f2 A+ Y
-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
! {, b! m, r+ F* Q2 ?6 C8 j1 L" S
! R0 X8 }: r1 r& k-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
5 L/ T2 a; W/ ^; Z4 [: V' F* ?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，
0 T' w( L- |" D1 j  f% o
& M' l! \2 d. U
: q8 A# u" w; X, F9 G
h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2
) N7 i4 S% m4 F3 i
7 d4 P! Y) g2 O+ z
6 n# J  a) D0 }2 m
" A; X' [/ |4 T-50时  个单位根                          N=200

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

. J$ s* x7 Q& c3 U/ z
F := QuadraticField(NextPrime(5));

KK := QuadraticField(7);KK;
1 G* t% a9 O& XK:=MaximalOrder(KK);
5 R6 z9 h  @: j# r' S4 X' ~Conductor(KK);
8 u, i* F0 t  J% jClassGroup(KK) ;
  f; e% S& u$ Q9 d- ]: `, g0 @QuadraticClassGroupTwoPart(KK) ;
) c+ f$ q8 V# t- J& u" q" ?% t8 fNormEquation(F, 7);
A:=K!7;A;
* f! L3 E. y* ~* ?+ ]2 zB:=K!14;B;
3 |$ L. o: F0 E# hDiscriminant(KK)

' O, P2 j, M3 C/ D  TQuadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
! w6 e3 h8 g8 H9 I3 }( Y28
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of K
Abelian Group isomorphic to Z/2
7 o+ `- W6 _7 l5 h# C, o2 {Defined on 1 generator
Relations:
/ k0 t/ m+ V2 b# {5 W    2*$.1 = 0
; t& S5 }$ x0 Q% x( f4 D4 dMapping from: Abelian Group isomorphic to Z/2
6 k* a+ }" ]7 e+ B1 \" e& v* ODefined on 1 generator
! Y0 c0 \9 j; z8 w8 ~# e/ kRelations:
    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
6 ^; w" A0 T1 g; ginverse]
# t: \7 ]% C* Gfalse
+ `  k( S* a6 g1 j; p% W3 J7
0 t% w; G1 d# `( H7 T3 n14
- q: p+ O' U! F" F. {% D28

lilianjie

# `( e& a6 X# G; n
1 s' n8 P8 t4 g& H. F9 R
" p/ ^5 [1 L  H
  J# o: H; o2 ?' C# |* P


6 r- f$ z& J$ X/ h
( j) K. Z* }# I0 C3 {分圆域：
- M' l3 g( p6 L5 `: t% c9 @& iC:=CyclotomicField(5);C;
! Z  h( J! [" T# T' m% QCyclotomicPolynomial(5);
C:=CyclotomicField(6);C;
' ^" |/ V! U3 d4 @) F) X7 ^CyclotomicPolynomial(6);
; |: |. N1 ]$ SCC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
, L+ N. O' \2 w* h4 tMinimalField(CC!7) ;
MinimalField(CC!8) ;
8 J" ^: Q5 P( O; B" z  E/ K: F) a) eMinimalField(CC!9) ;
MinimalCyclotomicField(CC!7) ;
6 B0 \! D; S/ xRootOfUnity(11);RootOfUnity(111);
Minimise(CC!123);
3 O/ d' q8 G6 a: p* g- N" lConductor(CC) ;
CyclotomicOrder(CC) ;
& T& K4 }  M% \+ t
CyclotomicAutomorphismGroup(CC) ;
7 ~3 F0 Y( Z! z$ J) E" p
Cyclotomic Field of order 5 and degree 4
) H) o9 f6 Q3 B8 N+ X$.1^4 + $.1^3 + $.1^2 + $.1 + 1
' l* l: x% \1 wCyclotomic Field of order 6 and degree 2
. q( O2 v8 x5 }; z5 ?( G9 E$.1^2 - $.1 + 1
8 G* K: z8 r# b: x8 K- ?Cyclotomic Field of order 7 and degree 6
$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
Rational Field
Rational Field
Rational Field
9 C2 Q% s+ U; L5 y2 M7 W5 Tzeta_11
4 ~( S* u" _- a! pzeta_111
0 l% ^& J% `2 ?6 v: J! }123
7
" D* C1 [8 V4 p$ A7
# {& Z- P& Y' TPermutation group acting on a set of cardinality 6
; m/ f) `3 M& k; k. d$ OOrder = 6 = 2 * 3
: A9 r; i) v; t    (1, 2)(3, 5)(4, 6)
/ L. m* m  h+ v7 f    (1, 3, 6, 2, 5, 4)
. J3 e# }+ J0 m6 ]# YMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
! W$ K' D3 Q1 a# v$ a6 F4 ~CC
Composition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
Degree 6, Order 2 * 3 and
& g) t- F1 |6 yMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
% G% b2 @1 k: Q' A6 @9 ECC

lilianjie

lilianjie 发表于 2012-1-9 20:44
分圆域：
C:=CyclotomicField(5);C;
* q; g4 @  D/ M( {9 `CyclotomicPolynomial(5);

分圆域：
分圆域：123

R.<x> = Q[]
+ x5 @$ B! s0 Z. x! \/ G: B: W5 ?F8 = factor(x^8 - 1)
F8
- ~( X- ?/ Y1 U6 M0 W8 }+ _
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

' {6 Z2 H* ^' }/ j: GQ<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
2 P7 i# b# A5 z- s3 oFF:=CyclotomicPolynomial(8);FF;
6 T: s; u# E* p- \& x" B
. B* N8 {' Y6 BF := QuadraticField(8);
F;
7 o2 @9 ~& X. X# C& hD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
! d: Q6 Z" c. `2 s$.1^4 + 1
3 G  f* ^$ V4 ]Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
3 O6 d2 Z$ d2 P8 \' R[
" |- j& @* L  k9 U5 l    <$.1^4 + 1, 1>
]

' c# h1 p+ v! ]6 {/ k3 e5 mR.<x> = QQ[]
F6 = factor(x^6 - 1)
- N! v5 ^) b$ i+ iF6

(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

Q<x> := QuadraticField(6);Q;
) ]1 c6 m8 N" l; j- lC:=CyclotomicField(6);C;
. Z, p% g, q6 }6 ^3 j+ EFF:=CyclotomicPolynomial(6);FF;
) y5 L5 J- ^; Q& P9 }1 K
/ L' O, ], [& E% S3 r- SF := QuadraticField(6);
$ K4 j+ E% \  N  J- YF;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
6 Z$ \, j! Z8 H5 |3 a# C+ J9 qCyclotomic Field of order 6 and degree 2
9 u2 Q/ {+ _, X. W+ q  H. x0 T$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
! S, a2 I' w- a  i, L9 G    <$.1^2 - $.1 + 1, 1>
& Z2 e3 Q& M2 H; @  Q]
( ^" N& O6 K. R" g
R.<x> = QQ[]
F5 = factor(x^10 - 1)
F5
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
6 m' E7 y* G. u1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
! i, a' T: K2 h8 }4 V
9 D0 s- E% g2 M8 |. iQ<x> := QuadraticField(10);Q;
! R6 w: U/ |# q& ]4 l: k  h7 f6 PC:=CyclotomicField(10);C;
8 }; X8 o5 N1 T% f" E$ hFF:=CyclotomicPolynomial(10);FF;
$ H4 r8 a5 G2 C: S3 @
F := QuadraticField(10);
F;
D:=Factorization(FF) ;D;
Quadratic 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 $.1^2 - 10 over the Rational Field
. S6 W2 l6 t7 F1 _0 W. h[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

lilianjie1

分圆域：123