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标题: 实二次域(5/50)例2 [打印本页]

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作者: lilianjie    时间: 2012-1-4 14:05
标题: 实二次域(5/50)例2

0 g9 x1 ^% f8 Q2 C4 \: @$ u4 D0 q. ]
- ?# q" `$ z# h1 M% {' c7 T( U: lQ5:=QuadraticField(5) ;
Q5;
Q<w> :=PolynomialRing(Q5);Q;
7 w$ F  q, `5 G: O: n9 i: v) w
, C7 T  l# Q# V0 G: dEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
' @( D' y% {6 F3 _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^2-3);
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
% p) j4 I( \" AFundamentalUnit(M);
  [1 Q7 x& B& O; [* M7 WConductor(Q5) ;
+ X6 j9 Y9 g" C  ]; P' A/ A5 C* gName(Q5, 1);
6 P  D8 c3 C/ q( pName(M, 1);
Conductor(M);
8 f" [. \# [  g5 O4 \ClassGroup(Q5) ;
5 _# @$ u6 W4 @ClassGroup(M);
ClassNumber(Q5) ;
9 m5 H4 l0 s3 a# c/ j. s/ JClassNumber(M) ;

PicardGroup(M) ;
PicardNumber(M) ;


9 }6 r' M+ F3 M' f7 sQuadraticClassGroupTwoPart(Q5);
! \3 V( Q* y4 A' uQuadraticClassGroupTwoPart(M);
# G( I' x" I# o/ a: m5 W( }' ]4 N

6 y4 z2 {+ y2 J+ M8 R4 \5 qNormEquation(Q5, 5) ;
NormEquation(M, 5) ;


" t9 I' n; F5 t, T" W% f  L2 YQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
5 x  [5 `6 C) C- ]# I4 q6 QMaximal Order of Q5
( [, j* h& s8 b% A' E7 ]Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
" v% B4 D1 ~  IOrder of conductor 625888888 in Q5
' z* G( N* r% Z6 O3 \0 N0 P" k3 \5 Ttrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
( ?$ \2 k! M: W; M9 K. _true Maximal Order of Q5
true Order of conductor 16 in Q5
true Order of conductor 625 in Q5
7 e8 |- }  d& z5 c; K- o5 O4 wtrue Order of conductor 391736900121876544 in Q5
[
    <w^2 - 3, 1>
]
. H1 n% w, T) ]- |# [5
1/2*(-Q5.1 + 1)
-$.2 + 1
& Y( ~, S1 o9 J; Y0 Q! G5
- L! q2 s0 W8 Q3 yQ5.1
$.2
. W0 B- j9 f3 G$ ^8 Y1
, F6 v  T, K, M5 F# Z+ _1 d2 SAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
2 o: T+ L+ A/ X5 iAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
! a$ u9 b1 ~5 @+ U/ [: Z( @5 C$ X1
Abelian Group of order 1
# \/ k( \3 n# 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
! e2 f- Y0 G) lMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
Abelian Group of order 1
. o( r: L  \; o# c$ HMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
: g; V( V2 P% ^/ ]. ztrue [ 1/2*(Q5.1 + 5) ]
: L- B7 |7 U/ f* @7 w$ ytrue [ -2*$.2 + 1 ]



3 q, y: S6 I) m- V$ Y
$ z4 X6 O. I3 `. z+ P* q2 h% U" f


$ Y, b# E/ [9 S; q7 H


0 Y- S% B7 a; {0 z
==============

Q5:=QuadraticField(50) ;
+ h  {. V) |& o( nQ5;

5 g% V! y! O' f; M& S& O2 UQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
" B; K. h5 z! b3 sM;
- v. @" E5 A2 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;
8 d, e# z0 ]0 _1 ]* ^( {IsQuadratic(Q5);
& A" V* q. A. i" i4 x( ~IsQuadratic(S1);
) u3 v' ^9 D& ]7 V  Y, @% qIsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
; `; \& f& l& J3 m, e  l5 JFactorization(w^2-50);  
  j. M4 b) W: `" cDiscriminant(Q5) ;
, a  p3 _( e8 Y& x" ?$ oFundamentalUnit(Q5) ;
FundamentalUnit(M);
& q+ a4 G  t5 I0 z9 JConductor(Q5) ;
! E7 C( W4 ~+ y  B
Name(M, 50);
Conductor(M);
$ g7 R" V# o2 I# h4 ~ClassGroup(Q5) ;
ClassGroup(M);
& N/ M6 ^9 |7 q$ |# m0 |9 W) d3 Z/ HClassNumber(Q5) ;
* T; M0 h: n' ?% n# _6 o! q, LClassNumber(M) ;
PicardGroup(M) ;
3 T* [! R) s+ X/ g: @* o% [PicardNumber(M) ;

* R" t( \4 Q; {3 m  s4 I" }% QQuadraticClassGroupTwoPart(Q5);
+ j' M# b0 G' `! x' X2 SQuadraticClassGroupTwoPart(M);
, o( c# `8 q; m$ W9 ?5 sNormEquation(Q5, 50) ;
! l1 V) n1 j) b: S; B% F9 l+ ^# Y* VNormEquation(M, 50) ;
- b: \2 d. O: l/ [  F+ m. @$ q1 A$ \
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
2 c& A9 c. m7 M7 t; |Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
3 l/ G. A$ W  J6 t8 HQuadratic 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
true Order of conductor 1 in Q5
; G3 P! x: ~7 L0 u" m/ X/ A( T( Ttrue Order of conductor 1 in Q5
" f4 z: ~. m: utrue Order of conductor 1 in Q5
$ I) S: n( Y+ x0 g3 g( v[
+ B) W% a1 p0 m9 ?' n% s+ r    <w - 5*Q5.1, 1>,
3 ]+ x7 Z2 m& i) \, ?    <w + 5*Q5.1, 1>
- D  V4 W' a$ m  y* k1 T]
8
1 j! E/ o& Y2 J+ t9 SQ5.1 + 1
+ s. p8 R! u! B- n+ P4 F: a, c- N$.2 + 1
8

6 r0 V8 h4 d* L" b4 U* W>> Name(M, 50);
9 o* w0 d7 b- O& d       ^
Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]

9 t6 Z7 p4 A3 T" H  Z; a1
Abelian Group of order 1
- y* a( A/ j9 G, h* Y, o' S+ XMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
; G$ a8 m1 h" F1 ]# T. wMapping from: Abelian Group of order 1 to Set of ideals of M
0 \$ E+ e' q9 z" T2 N/ W7 l! k1
8 K* b5 |. N& P8 ~1
" x- ?! C: W1 ~2 [Abelian Group of order 1
" ?! G* r2 Z( o9 ~Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
# E  f7 t2 |; t+ Y8 b; C1
7 v9 Z: F, [& M6 ^/ C( kAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
+ y$ H, x8 V$ a- X8 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
$ Y  ~$ g  e, H8 given by a rule [no inverse]
true [ 5*Q5.1 + 10 ]
' W4 g0 b' V" A; B8 Y2 A: }true [ -5*$.2 ]

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作者: lilianjie    时间: 2012-1-4 18:00
二次域上的分歧理论

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作者: lilianjie    时间: 2012-1-4 18:31


基本单位计算fundamentalunit ：
5 mod4 =1                                              50 mod 4=2

x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.
, ?6 B4 s6 d6 }/ d% J
4 |6 s& o( M2 A2 S1 }8 p8 R  l
( ]0 ?" d7 M- R最小整解（±2,±1）                              最小整解（±7,±1）
8 R2 W/ Y" t4 ?3 R. ^3 U' n. Q                                                             ±7 MOD2=1
. g$ O! S8 ?! G+ @
两个基本单位：

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作者: lilianjie1    时间: 2012-1-4 18:53

lilianjie 发表于 2012-1-4 18:31
基本单位fundamentalunit ：
5 mod4 =1                              50 mod 4=2

; x; e1 S/ F8 Q: o* X" M4 f1 Z/ W$ I基本单位fundamentalunit

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作者: lilianjie1    时间: 2012-1-4 19:07

1 Z. \5 H6 H! C1 A+ ^6 O' C& K
- H3 X# y& n0 [( h1 G) \2 D判别式计算Discriminant
- P! W+ C7 J- v7 G  {& _8 \
5MOD 4=1
) S. t- y1 H5 k5 V
（1+1）/2=1          （1-1）/2=0

D=5


8 R& b$ c9 o, f7 a: `( S50MOD 4=2
D=2*4=8

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作者: 孤寂冷逍遥    时间: 2012-1-5 08:37

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作者: lilianjie    时间: 2012-1-10 17:49

lilianjie 发表于 2012-1-9 20:44

分圆多项式总是原多项式因子：
5 a4 g8 L0 U! c1 w9 M# Q) wC:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

( H: t0 i* V$ {4 L1 T9 j; O7 y' Q分圆域：
分圆域：123
! w  E6 Y" P# q: K5 l
- W$ N8 Z/ p) |3 ]4 @( k1 ]R.<x> = Q[]
F8 = factor(x^8 - 1)
F8
# K3 I+ m* x+ J/ ]/ u' F
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

% P2 i- K" |8 IQ<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
) X/ R; h& z, p. O& ^FF:=CyclotomicPolynomial(8);FF;

F := QuadraticField(8);
: v# J6 w+ \. U% T: M) H5 kF;
D:=Factorization(FF) ;D;
% ]: z* \( e) NQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
    <$.1^4 + 1, 1>
# q! E6 R, `3 M! r]
' Z* p9 N0 |8 M4 h' A0 M. q
R.<x> = QQ[]
F6 = factor(x^6 - 1)
) j+ ^5 M! |) S6 l# ~3 f; M& b' lF6
. {; k6 T; O$ Y; A
( S3 v- g$ t1 o3 m% f(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

% f5 c: L! w0 B' p  `Q<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
" E+ g+ T7 u- s* rFF:=CyclotomicPolynomial(6);FF;
& d; S4 Q2 E0 U8 m% \
, A8 Y- K3 Z. \F := QuadraticField(6);
" X9 ~! K5 T3 J# @! GF;
' |$ i6 X) `6 I# }) wD:=Factorization(FF) ;D;
! g( B) t! X( S/ T* i4 zQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
5 R$ I9 u5 }: W; X. O+ |: F) ^0 R$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
4 j3 Y) z: O6 e! m    <$.1^2 - $.1 + 1, 1>
]
0 i' X3 f3 _% D2 S6 }
, F2 F% e' R5 P2 v# Y# Q% IR.<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 +
! F1 c& n# q8 ^/ ^2 Q/ x  C1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

4 T! B0 |1 U0 w- Y8 b0 O8 kQ<x> := QuadraticField(10);Q;
9 m4 z) e7 D' v2 w! tC:=CyclotomicField(10);C;
7 I/ j9 o2 ]: t; r3 _7 V1 c5 @; m7 lFF:=CyclotomicPolynomial(10);FF;

) [! O; E) Z( \F := QuadraticField(10);
0 V8 D9 [9 A0 R; XF;
' [7 s: T: Y8 S& @0 _4 e  v+ vD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
! x) J# ]( Y; [" ?! ZCyclotomic Field of order 10 and degree 4
$.1^4 - $.1^3 + $.1^2 - $.1 + 1
! d9 |6 D( ~4 J% m4 [: QQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
" F, `9 M$ |6 }6 J% R- t8 ^[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
# U$ b9 I# |7 ]( l, p0 D% N* \0 |* `]

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