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标题: 虚二次域例两（-5/50） [打印本页]

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作者: lilianjie    时间: 2012-1-4 17:41
标题: 虚二次域例两（-5/50）


- I4 o) y) N" eQ5:=QuadraticField(-5) ;
Q5;
( ^& t5 r. l' |, Z  ]
Q<w> :=PolynomialRing(Q5);Q;
( @9 ]( M2 R! k  ?  }6 R  `5 QEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
1 w" `7 F* }! a' l& n7 [5 R, V% LM;
# r2 {& E! q# n- L. I( Q9 ANumberField(M);
, y, a2 e7 ^5 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;
: ?' u' F6 g4 k! g" e, JIsQuadratic(Q5);
IsQuadratic(S1);
$ Z, V" e, {# h; sIsQuadratic(S4);
IsQuadratic(S25);
: k1 o3 y  T& w7 gIsQuadratic(S625888888);
9 l2 G, z- U1 P& A3 gFactorization(w^2+5);  
6 v" K6 b5 k. L0 K1 F- n+ h+ B6 B8 ?8 S, \Discriminant(Q5) ;
8 X. s  n4 t0 l- zFundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

Name(M, -5);
1 N9 I4 }$ z: y$ Y0 G. gConductor(M);
; I; o) ]" F! {& R! LClassGroup(Q5) ;
ClassGroup(M);
  j7 L" _4 \  u9 i0 b- S* sClassNumber(Q5) ;
ClassNumber(M) ;
! h% {2 U) `  i; a' dPicardGroup(M) ;
PicardNumber(M) ;
6 h, k; f0 M/ ^, A2 K6 H
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -5) ;
NormEquation(M, -5) ;
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
  b; f8 t# z; `' j$ |+ W2 WEquation Order of conductor 1 in Q5
- j+ J# j; ]- r  AMaximal Equation Order of Q5
( o# e& w0 R! j( b3 U! JQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
2 R1 t" j/ A% I3 R! |Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
6 F" ?+ q" [: |' Z8 strue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
$ {/ v& _% m% K& o! D" K& u7 M- ktrue Order of conductor 1 in Q5
1 z/ r; b$ c4 Jtrue Order of conductor 1 in Q5
[
/ v3 l. ^+ \! W$ s5 @9 [    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
-20

>> FundamentalUnit(Q5) ;
! q, p$ J" t% X% m6 n& C* c                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant


4 A$ q  I0 I% U3 B# W# ], Y2 l>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

) S! E; K" P% c, t, P20

>> Name(M, -5);
/ s" v, l0 n( N/ B' g       ^
% ~. O& N! Y. D3 _$ W9 aRuntime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]

$ C3 e* O) D1 G1
Abelian Group isomorphic to Z/2
Defined on 1 generator
* c; F" V* @% x( c4 r. _+ `Relations:
7 f: A- {; c+ \# K4 H, s    2*$.1 = 0
7 v, \6 I# |* C$ ], D- R- F7 eMapping from: Abelian Group isomorphic to Z/2
9 N5 Q1 l. \8 n& z! ?1 K! XDefined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M
/ j7 A5 i8 a' g! G' y2 |Abelian Group isomorphic to Z/2
$ n1 n- u2 n! s& qDefined on 1 generator
( a1 Y3 d# i# {, U3 yRelations:
# z0 i: z5 B7 j3 {& y5 t2 d    2*$.1 = 0
/ u9 Y6 O# @9 p. Y( @' z. B5 tMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
3 E( S0 @/ v3 k. k; C* m( YRelations:
    2*$.1 = 0 to Set of ideals of M
. I7 D% W9 A( c$ a* B% ~4 f2
2
; f/ c4 ]: @7 `" J7 X% s4 A0 VAbelian Group isomorphic to Z/2
" K  {  T* U' h# c; {7 oDefined on 1 generator
Relations:
- }" k; H. [2 e) I0 t( r    2*$.1 = 0
. U% a" g7 X& T  w0 ]Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
+ X" d/ D3 D# f" xRelations:
    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
2
$ v+ a2 Q5 \# S0 P" y# YAbelian Group isomorphic to Z/2
Defined on 1 generator
* [& b+ {, D. r0 u# \' |* ~3 ERelations:
$ x" y1 k( E' q  [# u; o+ K    2*$.1 = 0
! d/ ^1 e( k1 E- VMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
9 U$ m8 z' ?% _) Z    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
6 W0 B! g5 F0 X9 L0 @- u9 N) ninverse]
4 |1 U. A6 r- u% ]) H( s. HAbelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
  m, ], B4 ^/ h" a+ s' o4 \Mapping from: Abelian Group isomorphic to Z/2
6 ^+ G, Y0 {% N# {  {7 Q. mDefined on 1 generator
+ J( X7 Z( G9 f/ m, m9 |7 R2 LRelations:
; t, n0 A1 ^" k' @2 C2 _" |3 r    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
' k( v0 g8 K! T/ einverse]
false
false
, K2 _+ b" z% F6 M==============

4 O  E5 X/ `  l+ w
/ b2 Y5 }: f5 X, _' m4 MQ5:=QuadraticField(-50) ;
4 R; f2 I' X1 P& |7 I- Z+ NQ5;

Q<w> :=PolynomialRing(Q5);Q;
- |7 B1 ]+ g8 ]2 e. H- ]0 e3 J" u  cEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
- ?: H/ z$ W7 S: eM;
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;
' X) Y6 j$ Z% D: @IsQuadratic(Q5);
$ J" n4 d/ h8 K6 H  GIsQuadratic(S1);
IsQuadratic(S4);
  ?/ n7 X/ U/ h4 @8 N( _# wIsQuadratic(S25);
IsQuadratic(S625888888);
$ K! |* G1 [7 U2 \7 ZFactorization(w^2+50);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

Name(M, -50);
( r3 L6 U  Q4 I9 D. X2 @Conductor(M);
ClassGroup(Q5) ;
/ v7 k1 |2 H, j( v+ N: z- PClassGroup(M);
ClassNumber(Q5) ;
1 x4 ], \4 u8 i% B+ UClassNumber(M) ;
4 [8 M& h/ X) I- VPicardGroup(M) ;
PicardNumber(M) ;
3 J: a" t) j2 o$ p, D4 P
: q3 M3 W, A4 B6 gQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
- f# n, L+ i* T( Y6 XNormEquation(Q5, -50) ;
. P/ ]  B, h0 T$ q5 GNormEquation(M, -50) ;

; F- `: }6 {+ zQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
/ y6 m2 F/ v2 p# R& J' r/ OEquation Order of conductor 1 in Q5
' E- \/ v8 X# S4 N  dMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
" X- s0 q) r# oOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
0 y" R  t6 S0 I% p% ~3 ]true Maximal Equation Order of Q5
( A9 D1 Z& Y' d" u$ g) B. P! atrue Order of conductor 1 in Q5
: o3 I8 ^0 N2 \8 Q4 t6 e4 Ltrue Order of conductor 1 in Q5
; E  G4 ~7 z6 A; b, utrue Order of conductor 1 in Q5
1 X2 k/ W$ R2 u$ p  e9 i5 m* E[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
-8
: i* E) R; D/ E7 V% }9 ]
* `; z+ p: p4 n- [+ K>> FundamentalUnit(Q5) ;
                  ^
: b$ q& H* a7 PRuntime error in 'FundamentalUnit': Field must have positive discriminant
8 }, v7 A" `" B+ K; m  E3 H

* N' J7 h% A$ ^0 o) n7 k>> FundamentalUnit(M);
                  ^
" q6 A0 r6 K  H, g% N4 C, qRuntime error in 'FundamentalUnit': Field must have positive discriminant

8
& E5 _4 h5 ]$ Z! Y; U
8 y& f9 X) @* E$ ]! ^>> Name(M, -50);
/ P; ]& K* {) j# N       ^
: ?/ A0 m. i: l4 n. yRuntime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]
; E2 O* o8 f0 @8 s' s# m
1
, V1 q+ {; o8 P! CAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
9 s2 m/ U5 ~: s: G, W0 Q: qAbelian Group of order 1
' e" v$ p6 b, m' z5 b, b  R9 W/ LMapping from: Abelian Group of order 1 to Set of ideals of M
1
9 m* |4 G$ ]) M% I0 a/ |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]
3 B( X" B& ]/ ?, L  V1
4 `$ `: g( y4 X" ~2 r1 ~Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
& S$ A# x' x9 M3 n  P-8 given by a rule [no inverse]
6 r4 @  C- [3 nAbelian Group of order 1
" L; `  r  K; k5 fMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
$ I+ `, {7 H9 Z2 q-8 given by a rule [no inverse]
& ~# K$ H/ r) Q8 D, o, }; ~false
; a( c( z0 v9 |* S, yfalse
" I# S8 s* v) u7 i* Z3 z) v5 A

---

作者: lilianjie    时间: 2012-1-4 17:51
看看-1.-3的两种：

# n0 C# K2 K9 E; U- g' }" AQ5:=QuadraticField(-1) ;
Q5;
# j* F: T0 G+ Y# f. G" \5 a0 N: Q
) g' k: E0 O3 M( E: i$ T) \# Y% {Q<w> :=PolynomialRing(Q5);Q;
& Y% ^" |! ~. q4 |1 i( uEquationOrder(Q5);
& x" j' ~  |. M# {$ v3 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;
: \' v! P6 [4 J. T5 e. |IsQuadratic(Q5);
% `+ |* ~4 l& WIsQuadratic(S1);
IsQuadratic(S4);
- C& Q0 k$ @7 W. P' V; @IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+1);  
Discriminant(Q5) ;
" B4 G5 a! u2 B* R9 Z  Y5 G6 pFundamentalUnit(Q5) ;
& D5 N+ o- r0 D' s+ [FundamentalUnit(M);
Conductor(Q5) ;

Name(M, -1);
; T* e: R6 ^, W5 ]- I& I; mConductor(M);
ClassGroup(Q5) ;
0 K5 W( l( ?6 CClassGroup(M);
) p- Q/ j$ |( K. VClassNumber(Q5) ;
) _/ W! F, x) t( c/ q8 ]7 d$ A7 pClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
0 g1 s3 t( w9 k( N  u% \  ]
  ~" Z2 v' F, j  f: mQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -1) ;
NormEquation(M, -1) ;
  H  a7 T/ G* I/ r/ c( F  F! A
) [8 e2 u: b' g$ z, JQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Univariate Polynomial Ring in w over Q5
6 w4 O# ^" P$ Z6 S) EEquation Order of conductor 1 in Q5
" ?/ h, b/ v- t7 p7 x9 K* h9 RMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
& ^/ s3 t" Q, H% Y: t: q& lOrder of conductor 625888888 in Q5
4 X, V; L" S: N9 a3 D8 F1 B4 Ytrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
true Maximal Equation Order of Q5
- I. _) N& U8 A" z/ T2 d. ]  a1 x  |true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
' I8 s- Y) t2 @; p* {9 T$ otrue Order of conductor 1 in Q5
9 U7 w+ ^# s! p' z- P[
( g+ z; W7 d" I) ~* S7 A7 q! t7 k    <w - Q5.1, 1>,
    <w + Q5.1, 1>
. k3 Q/ D* ?2 o) n$ ^3 P$ g]
-4
# m% K# C) k( U9 j3 I9 u
>> FundamentalUnit(Q5) ;
* z! F  Y- r$ [# F, A& |                  ^
8 }+ t6 P" C+ ^. g" C( \* r8 dRuntime error in 'FundamentalUnit': Field must have positive discriminant


5 ?2 o8 s% K% p+ Z, C, ]1 J>> FundamentalUnit(M);
                  ^
  V# s; |, P& w/ O6 JRuntime error in 'FundamentalUnit': Field must have positive discriminant

" t$ v! E0 a4 Q& v  G) |4

>> Name(M, -1);
       ^
" ~0 _+ A2 Y$ x. p! TRuntime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]
* {+ R3 m" k. v3 E( z5 l
7 q2 X1 n1 R8 ^5 H) n# a1
1 H. c: E4 a, y) o( E% oAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
9 v7 w% o' ]& g: C: ZAbelian Group of order 1
+ W* ]- }# h: U, H9 @Mapping from: Abelian Group of order 1 to Set of ideals of M
8 {. w5 O$ Y  C7 z1
# o2 {3 v! ]9 r  f. P& x1
: u$ o! S' K) _. OAbelian Group of order 1
1 ~6 K! p0 c! U5 N; n7 pMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
& Q7 E" `% p8 e* q7 U( f: a" Uinverse]
* H0 ]& P& z2 E/ ^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]
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]
8 ^/ e) t0 Q# c, ~, Jfalse
false
. \4 J5 F5 u+ W" ~- k. d0 H===============

Q5:=QuadraticField(-3) ;
Q5;
9 }% l* ?, i% [& x. |" _
9 i. X6 l3 @  u" D9 S; WQ<w> :=PolynomialRing(Q5);Q;
& @: `% n8 W8 t' J  sEquationOrder(Q5);
+ j& H& m  R2 XM:=MaximalOrder(Q5) ;
) a2 f& U8 s7 R. e0 {M;
4 x/ w  w1 a6 }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 p3 F+ R8 `; n: t: P; w7 OIsQuadratic(Q5);
. m1 G; n0 q. p* D. k) U  zIsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
6 Z: J. b" s) Q2 ]! WIsQuadratic(S625888888);
Factorization(w^2+3);  
; B# O% [" k, q6 s" X3 KDiscriminant(Q5) ;
" T/ f8 H2 P( B4 e' }FundamentalUnit(Q5) ;
6 I4 a: B$ \2 NFundamentalUnit(M);
  q: h4 n' x6 [) y4 [( o1 W( H: jConductor(Q5) ;

5 m! s) n, ~) P7 R5 B6 DName(M, -3);
) t! }* c  t5 S8 H6 bConductor(M);
ClassGroup(Q5) ;
3 H- k* P: x' j3 T( z2 AClassGroup(M);
( w8 S. O) q8 _( u" sClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
1 q9 Q/ l5 ?2 @! f; OPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
7 v1 v# e! Z8 ^' v* V5 B+ bQuadraticClassGroupTwoPart(M);
NormEquation(Q5, -3) ;
NormEquation(M, -3) ;
/ y0 a7 l) x' M
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
0 G$ g" ]: {/ T) J9 fUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
Maximal 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
* h3 c: t1 a, c7 Ptrue Maximal Order of Q5
# N7 _( U: G3 I& ~4 `true Order of conductor 16 in Q5
true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
% W( _: V% J5 p. a7 q2 O& @-3

0 o5 C0 w3 ^  M3 f) z0 A>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

! {' L7 G9 f- U3 Z. e$ ^9 J
>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

# U& T! I) _8 r* M3
( W6 u" e; t/ z, R( w1 E
1 v% v. s* U0 S9 D0 U# G8 _9 Q+ F* X>> Name(M, -3);
       ^
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]

5 _' i1 d1 b8 e2 M5 @# |1
! C* J8 k( y" l+ l7 XAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
# _" t  [; V8 w" G3 HMapping from: Abelian Group of order 1 to Set of ideals of M
1
1
Abelian Group of order 1
( Y. N& O0 w  yMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
5 E4 }' ^; l9 a" b1 u1
2 p+ m2 M# p8 `7 }( r) cAbelian Group of order 1
0 d3 d/ {! c5 S; |+ v# AMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
$ B5 Q/ W: a, E! Z9 s- c5 ]-3 given by a rule [no inverse]
6 [& a' S# b# L, J( c$ ?) W7 _1 _Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
3 l2 S, W0 G  t- C: K1 J9 ~0 L-3 given by a rule [no inverse]
false
9 m( {4 P! M1 ~9 s$ Cfalse

---

作者: 孤寂冷逍遥    时间: 2012-1-5 08:36

---

作者: lilianjie    时间: 2012-1-5 13:02

$ V0 q1 v. Y4 N5 |' i
1 `) j' A! }9 _: r% P" FDirichlet character
, r, t8 t, e9 m9 B$ cDirichlet class number formula
0 K$ A7 |9 Q# J
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
) Z' }! Y6 t. s
* F5 p4 ]: ]: _! Y3 D5 A-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

( u4 J( @% o% o6 S-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，

  M0 F% Z# t: x: B$ E# G

" e$ M$ `( E9 O* c% ]  z) Eh=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2

9 G6 f8 A% `) d, ?9 s% y" l/ l

-50时  个单位根                          N=200

---

作者: lilianjie1    时间: 2012-1-5 13:51
Dirichlet character

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作者: lilianjie1    时间: 2012-1-5 20:37
pell   equation

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作者: lilianjie    时间: 2012-1-9 20:28


F := QuadraticField(NextPrime(5));

; _5 u$ u- h3 I) l3 t0 v6 x" _KK := QuadraticField(7);KK;
! l  d# ?7 _+ x0 x& `- d& H' xK:=MaximalOrder(KK);
0 w& o* g- ^* M' f2 }Conductor(KK);
$ C5 |& c  G4 }/ |  NClassGroup(KK) ;
QuadraticClassGroupTwoPart(KK) ;
$ a* \+ Q* W* z" O; `7 QNormEquation(F, 7);
A:=K!7;A;
B:=K!14;B;
Discriminant(KK)

Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
$ @- q* _, V0 F/ S: v28
+ F2 o- @" t. [  T. E0 y+ g1 {Abelian 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
' i: g0 i) a' m! M& zRelations:
    2*$.1 = 0
3 p0 c9 h+ C5 U/ Z$ IMapping from: Abelian Group isomorphic to Z/2
  N( @; q, L) r/ U6 O7 {Defined on 1 generator
6 K/ x3 A0 ]9 ~/ @5 u" L7 ZRelations:
    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
inverse]
false
7 B9 S4 E  @7 B  x) T0 q7
14
* T  g' e) C# W28

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作者: lilianjie    时间: 2012-1-9 20:44

) [6 Q! j. x. B- n6 i
* t( _) X( @- a) h. s

1 p$ {! `  J% F8 w" k
5 e1 c) Q! U! {: y8 E
/ j. ]! k: X: o0 x4 V% f0 Z0 S

1 o$ h0 @( d2 h. e. }分圆域：
C:=CyclotomicField(5);C;
. J$ p# b5 p3 u- }4 x) kCyclotomicPolynomial(5);
C:=CyclotomicField(6);C;
# A% Z/ ~/ o$ J6 y, s# O6 h, W  Z8 ~CyclotomicPolynomial(6);
CC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
" a3 X/ b5 `' q& e, xMinimalField(CC!7) ;
MinimalField(CC!8) ;
/ e8 K& P& H3 |, U0 r* n. K+ _: [MinimalField(CC!9) ;
MinimalCyclotomicField(CC!7) ;
& W0 O; I: p, g  i* W/ rRootOfUnity(11);RootOfUnity(111);
& g+ D4 [. T! g' O5 l- X6 G$ iMinimise(CC!123);
: D8 N0 C$ {( s( B% bConductor(CC) ;
- P0 y: F  E  L  R8 J6 s' pCyclotomicOrder(CC) ;

CyclotomicAutomorphismGroup(CC) ;
0 g+ ^0 d& x" k: U
Cyclotomic Field of order 5 and degree 4
* i5 t! d0 `' Q$.1^4 + $.1^3 + $.1^2 + $.1 + 1
Cyclotomic Field of order 6 and degree 2
! s( W( i) {( i( o+ V$ z7 w$.1^2 - $.1 + 1
Cyclotomic Field of order 7 and degree 6
7 Q% ~. j) Q5 b2 U# u) A$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
8 M6 z3 |* O" W. uRational Field
. O  v5 n# {6 b* ORational Field
( M$ P: z( R& o6 S5 I/ vRational Field
Rational Field
zeta_11
" @" T7 r$ Q: J+ Czeta_111
. O) H* y. Y" }( t) X. s123
7
7
0 }8 u: f$ F$ z& Y5 R; L3 Q- HPermutation group acting on a set of cardinality 6
Order = 6 = 2 * 3
+ D0 q* D# z! Q6 u7 ^7 f0 l    (1, 2)(3, 5)(4, 6)
7 M( z! E3 ~* h* J    (1, 3, 6, 2, 5, 4)
* T3 [- p) ^) P* p1 T# PMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC
: i6 A3 K) R$ J' W9 c8 Y6 VComposition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
Degree 6, Order 2 * 3 and
# c  z# A8 ]; e9 ~9 i2 iMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC

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作者: lilianjie    时间: 2012-1-10 11:34

$ V- F% x* H6 f% D

lilianjie 发表于 2012-1-9 20:44
分圆域：
4 g) i4 G+ l, G& s0 G  K0 ~7 k3 b( SC:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

, l6 W) X# l% z3 X( A* O- y( }
分圆域：
" o5 H) u+ D- O( P分圆域：123
$ z' f- H9 i! u' {. x, S
R.<x> = Q[]
% H. |& y- i: P0 k, L/ X' h4 CF8 = 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)

9 u' A7 D/ Z$ D' D" ]2 H/ LQ<x> := QuadraticField(8);Q;
$ r, y8 A! f7 z) ]; {C:=CyclotomicField(8);C;
2 H" J; m7 E  o/ y8 @+ EFF:=CyclotomicPolynomial(8);FF;

F := QuadraticField(8);
5 N0 v& o& ^& U; yF;
8 s+ p# B8 C3 BD:=Factorization(FF) ;D;
+ T) O! V' t; `) ~2 e3 PQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
+ H% M4 E' x, `3 K) {! r- J: ]' c$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
9 m9 w0 G5 K$ o[
1 o" p  o, C1 z0 C3 V0 h! Y! n    <$.1^4 + 1, 1>
]

R.<x> = QQ[]
; ?/ @- l6 V8 ]% JF6 = factor(x^6 - 1)
F6

) J/ A# ]9 g/ f6 i. W9 v9 m" S(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

( }/ p; y, K% jQ<x> := QuadraticField(6);Q;
: {' Q- V' J, j; ]$ V7 ?C:=CyclotomicField(6);C;
2 @/ R) [2 O* H/ lFF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
F;
8 u5 q5 J# [' ?- K. g. GD:=Factorization(FF) ;D;
! |5 y9 N* n8 i# O4 F, _& a3 O; uQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
4 h1 J' _. G) K! RCyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
0 b8 }, k1 w* ^! F0 n( F) l    <$.1^2 - $.1 + 1, 1>
]

R.<x> = QQ[]
F5 = factor(x^10 - 1)
  O3 t, Q$ ~" f1 M- bF5
7 I- e! Q- m0 O( g5 M& Z(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
  C* w# G- y9 e& p) ^1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

Q<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

F := QuadraticField(10);
F;
1 u2 E5 ^8 X* X" ^7 I- C" `( [D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
) i" V& l( k  |/ I1 W, M) D2 Q$.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>
]

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作者: lilianjie1    时间: 2012-1-10 11:55
分圆域：123

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