实二次域(5/50)例2

lilianjie

Q5:=QuadraticField(5) ;
5 q0 x9 {$ F) u$ w) `Q5;
- L) W" V4 n+ k1 y4 MQ<w> :=PolynomialRing(Q5);Q;

- Q. ~, E( M' f5 w) VEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
. ~, d' R; j  E1 {NumberField(M);
2 C2 @: q: u) n$ V, eS1:=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);
' E' \' Y" j/ CDiscriminant(Q5) ;
/ X* b; E: D4 k: p5 ^8 YFundamentalUnit(Q5) ;
FundamentalUnit(M);
6 G0 |6 Z2 f2 T& L$ _0 xConductor(Q5) ;
Name(Q5, 1);
Name(M, 1);
Conductor(M);
6 `6 B2 S% Y) H* a6 {ClassGroup(Q5) ;
; A/ o- x# I0 NClassGroup(M);
: c4 v7 o( a, V: B9 O, `ClassNumber(Q5) ;
( t/ R/ A. C) n, z, E/ Y) eClassNumber(M) ;
3 |  G# t4 ^' Y
PicardGroup(M) ;
- p" C, ?# @2 Q' }& X  ?5 QPicardNumber(M) ;


! g$ k* B2 y7 x& G. l8 p) f3 lQuadraticClassGroupTwoPart(Q5);
4 u5 b: m: @5 G/ U9 q- Y  k+ ^. F+ gQuadraticClassGroupTwoPart(M);

. K! M; \5 a4 r% t/ s. r. M0 ^6 w2 U
NormEquation(Q5, 5) ;
+ P6 D! P* b) b  E) QNormEquation(M, 5) ;
$ B5 x  R. o- r6 u4 x0 g6 G1 t( Y

Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
& A# e, A2 Z, J' z0 Z1 d4 Q$ N, m8 ~Univariate Polynomial Ring in w over Q5
4 Y, L* ]2 H/ ~( q7 d9 m# IEquation Order of conductor 2 in Q5
Maximal Order of Q5
  m6 m0 T/ ~% U/ N5 v: |1 LQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
0 Q* `8 }& ]" L" }: pOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
. D7 ]" v: v6 f: k) htrue Maximal Order of Q5
true Order of conductor 16 in Q5
true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
[
    <w^2 - 3, 1>
! v0 ^0 J) k! |) t5 p]
- L% I6 Q% p/ ]5
2 n- g: D0 a/ E7 x7 [/ ]& _1/2*(-Q5.1 + 1)
" {9 s- X* J- e2 a/ V/ ~) w-$.2 + 1
8 y! y* r% G2 m" l2 h' D5
  r5 T3 q. j6 J" P$ |Q5.1
$.2
1
) ^7 f' |5 R, |* z4 ~Abelian Group of order 1
8 ^# X: K* M5 Y% s8 N9 r+ EMapping from: Abelian Group of order 1 to Set of ideals of M
  Y5 W) a: }1 v8 r7 [( F9 EAbelian Group of order 1
: M# @& F# l, j  T. \Mapping from: Abelian Group of order 1 to Set of ideals of M
1
; ^; S, Z8 E; _, w; T1 _1
( |# a# ~, r6 T; z/ _+ A( HAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
  w4 b6 }; K9 |, i9 l) Qinverse]
7 {) E, X* b3 O0 C1
7 r4 c8 |- b# T+ v, FAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
7 {' X+ S/ ~4 k1 P* V5 given by a rule [no inverse]
Abelian Group of order 1
' Y  c2 J) O- gMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
  ~1 l" _* k( l+ f8 @9 o. x5 given by a rule [no inverse]
true [ 1/2*(Q5.1 + 5) ]
true [ -2*$.2 + 1 ]
& r# G# C8 T( Z9 p



, k1 a) @! U9 m& A0 ?( H
) A) ]! D1 [" G" H- A+ n
) l/ Y. B9 X% o* _, G& ?) Y; x
& M1 \, l. x; ?2 O

! Z' l+ x) ]: f2 `! J3 t3 H
3 D1 e8 w2 V0 E+ D9 T% i. z
==============

Q5:=QuadraticField(50) ;
Q5;
, v: K6 c7 T5 }/ o# v; }
Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
% {; A5 t8 q9 u; w, g4 R1 G6 v# WM;
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;
5 |# I$ J0 ^- v& ~IsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
8 r5 a1 N$ d$ a: q! i8 \IsQuadratic(S25);
IsQuadratic(S625888888);
; b  i! O+ r1 \$ ?, j# kFactorization(w^2-50);  
4 E7 C- r/ `/ E) I9 X0 }8 {" C# FDiscriminant(Q5) ;
: a% g1 P$ ?& ^3 N0 E# KFundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

Name(M, 50);
Conductor(M);
2 ^  O& ]& C* _& nClassGroup(Q5) ;
ClassGroup(M);
) v1 U* b& c: n# W3 JClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
. {# j' c; x0 \* s& sPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, 50) ;
NormEquation(M, 50) ;
( A# F+ m- t' ^# T) d
2 P) ]- R4 Y, ]5 J/ z7 LQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
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
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
- V; v* ^1 B4 p6 ktrue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
& ~; L/ ]4 s# D2 X6 S" xtrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
! o& s0 ~3 p( Q3 d( z[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
8 R  I$ i0 }% ]8
, q* W: Y# ^& [+ ~! m! DQ5.1 + 1
( F# z# [* [( g3 f) z* [$.2 + 1
/ z: S# W* V9 h( e9 Z2 {; m2 D0 j' r( o9 S8

* ~( w) L1 ~" ^# W9 M3 y) P>> Name(M, 50);
: Y/ p' A: T$ k) i: c$ ?; A       ^
3 S" d- \* C# M- Z" x) FRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
3 X  x/ a$ {! u- V$ o
4 k/ D6 @# S  u& ]' D! @2 R1
Abelian Group of order 1
+ g5 D1 h0 h% s# PMapping from: Abelian Group of order 1 to Set of ideals of M
/ l0 [7 ~* M0 X/ n& Q1 _# JAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
1
Abelian Group of order 1
1 [, w4 c8 \# m' [" d" k8 `Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
" S3 ^% _: J$ P  \' Iinverse]
0 {$ x9 @$ B1 Z+ X% q1
; T5 p, L9 G7 {; x2 ^9 y6 l% kAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
% I% e) d6 r/ U: m  `2 dAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
- x# m0 _! `# q( z# _8 given by a rule [no inverse]
true [ 5*Q5.1 + 10 ]
" E/ T! q: p. A$ n& H. U& jtrue [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

# B- a- y, {" n2 ?
基本单位计算fundamentalunit ：
; l# }9 U* B: B+ g2 l5 mod4 =1                                              50 mod 4=2

$ h" G. k. w& w- A7 }& Q! Q/ r x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
1 i6 c+ U5 H5 F2 W$ F x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.


6 H  o( N; a; ?! V1 }: E最小整解（±2,±1）                              最小整解（±7,±1）
                                                             ±7 MOD2=1
: S1 P6 Z7 o( ]6 s; G9 O# L& Q( K2 h
两个基本单位：

lilianjie1

lilianjie 发表于 2012-1-4 18:31
- J& b7 q8 |% `- N基本单位fundamentalunit ：
5 mod4 =1                              50 mod 4=2

基本单位fundamentalunit

lilianjie1

: s. a& R6 F( R6 z6 o3 o% T. d
: ]- t$ K) d3 B/ P9 `4 |判别式计算Discriminant
! `3 m4 F  n5 p8 A7 t3 [5 F, C' N: Z0 T0 O
5MOD 4=1
: w0 V& d8 S  V
（1+1）/2=1          （1-1）/2=0

$ y8 [, I( M# L+ \D=5

6 j7 H2 @5 k- N% R& ^/ c# g
/ |% |3 S; ?2 q$ |1 R7 \50MOD 4=2
) J1 B$ B. u) X; fD=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44

分圆多项式总是原多项式因子：
C:=CyclotomicField(5);C;
: R$ S' m' o2 c# L" JCyclotomicPolynomial(5);

+ S) H2 F  s/ [9 H6 k
2 ]1 x9 v8 J7 Z$ t* V分圆域：
分圆域：123

& U1 c4 x$ L" d/ @5 o& x1 eR.<x> = Q[]
F8 = factor(x^8 - 1)
F8
% A4 ]2 w5 p: |% a4 o
(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;
C:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;
# N6 G( m9 f) D& I- g9 Z
% h0 ]) x3 J8 G0 j% T+ L$ Q( bF := QuadraticField(8);
F;
2 K. I) X$ S5 b, w( l6 FD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
. U4 _9 h9 r+ ?  p2 u2 y3 q- K' T$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
1 }3 I7 |0 B  s6 M* n    <$.1^4 + 1, 1>
]
; ~3 H  c( N/ B
  D  H" ]6 k3 O# WR.<x> = QQ[]
F6 = factor(x^6 - 1)
F6
- l( e. @' h4 d) i; R; J
- s, i2 a( ]$ a1 _5 b  l8 {* Q(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
+ u' \" i( s3 L0 m5 Y. z
2 [7 o7 N7 F+ d5 q% @% `: f$ R4 DQ<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
8 D, ?  j1 R0 _& y4 B& FFF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
, @0 z1 b5 Z& r1 `, X0 LF;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
8 M1 g1 w( C+ [! @0 }6 y& ~Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
    <$.1^2 - $.1 + 1, 1>
]
' u2 J7 i! c7 s+ q5 A" j
R.<x> = QQ[]
4 ~- l4 P1 d# A* m% T/ ~# EF5 = factor(x^10 - 1)
F5
( q7 r, e3 M9 p" D(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
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;
6 h  }1 v/ ~' ?1 P7 A) [- U, ^C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

0 u+ z5 l3 }4 eF := QuadraticField(10);
# P. v6 \( A" x0 }( W/ k( g* ^F;
8 A3 |3 w( W. u0 y+ QD:=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 $.1^2 - 10 over the Rational Field
& D& C( Z- D! U9 j2 ][
9 a) E+ e) G/ }, e* i6 Q) k    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]