实二次域(5/50)例2

lilianjie

7 |: r, w0 [5 \4 W' S9 KQ5:=QuadraticField(5) ;
Q5;
Q<w> :=PolynomialRing(Q5);Q;
. P, @3 u/ x4 F- d
) c6 k0 k) h0 \- bEquationOrder(Q5);
2 B* |& H! G- C; F# k7 M% YM:=MaximalOrder(Q5) ;
M;
NumberField(M);
3 g6 ]; `8 `4 E7 hS1:=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);
  f: @, N2 o' I  [6 L6 J7 W8 bFactorization(w^2-3);
Discriminant(Q5) ;
2 D2 J/ }1 i: s. mFundamentalUnit(Q5) ;
# x( T9 O) n7 K  y# yFundamentalUnit(M);
4 D7 o7 _" X; @/ FConductor(Q5) ;
- B" b4 ^! }" o- G1 _2 _Name(Q5, 1);
- X& v3 K$ q5 g* k! a' y# DName(M, 1);
Conductor(M);
! e6 h8 q- O- ]5 L9 W5 C# aClassGroup(Q5) ;
8 Y% s1 k/ X* q3 YClassGroup(M);
, L  K5 f) P! w+ _) l7 JClassNumber(Q5) ;
ClassNumber(M) ;

PicardGroup(M) ;
# D$ @& C& e$ Y! iPicardNumber(M) ;
% Y5 D4 N" f' @, P

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);

3 S1 u) Q' F4 d& g! }1 `3 q
! K: c% U# _% U3 R! x* aNormEquation(Q5, 5) ;
NormEquation(M, 5) ;
9 ]- P3 D% v  a, k+ A. V+ A

Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
8 h1 S* R, t+ ]8 Q* Z7 R$ uUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
0 l7 B  a5 z" F# A# b4 d/ fMaximal Order of Q5
( j; `" o5 P0 o4 y; l* l$ ^Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Order of conductor 625888888 in Q5
  ]& R. o$ E9 a1 h+ [1 mtrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
5 i# q- S& u/ m; K2 R! Ztrue Maximal Order of Q5
true Order of conductor 16 in Q5
% }9 [- _+ B% Y1 ?true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
7 b5 S/ d$ ~$ d/ S3 ^$ j[
. p) G* ^! R: F  m9 A3 n    <w^2 - 3, 1>
]
5
% b: W7 Z! R& W1 ~* g( D1/2*(-Q5.1 + 1)
) V7 U) N2 N& ]2 Y; f6 u-$.2 + 1
. q# k6 B- x0 l" p5 m: `8 Z5
# f3 V2 l0 M. }7 U7 h  Y4 \9 iQ5.1
( w+ p. F& a4 g$.2
1
% [# x+ @  h% r" }7 b) OAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
6 b; C" d2 t2 c8 r3 _- RAbelian Group of order 1
+ r7 p% o- v+ }5 ^/ `4 fMapping from: Abelian Group of order 1 to Set of ideals of M
1
" k# n6 J2 k# P! M0 ^1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
8 r6 u( C" Z" R% R' @+ z( n' u' linverse]
1
Abelian Group of order 1
; P# n: V9 @) }( N/ O6 XMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
/ x- A& g- A- e" y% ?$ T' Y5 given by a rule [no inverse]
: I) a& M, d% g- c  M2 ]& a3 [$ wAbelian Group of order 1
: ^* r. C/ j5 ^( S- R0 iMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
: u. C" Q; {8 W. Q) z9 t5 given by a rule [no inverse]
true [ 1/2*(Q5.1 + 5) ]
true [ -2*$.2 + 1 ]
8 H# `+ @0 Z! \+ A  y

. i( j; I* N( Y- I: F


$ P: B& @8 S# S9 m  d! G5 U/ F3 L

1 i+ W6 j5 B; f; }0 g



==============

Q5:=QuadraticField(50) ;
Q5;

9 f* ]' m8 K6 R+ BQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
. v+ S  G' q6 I% H. eM:=MaximalOrder(Q5) ;
& F! o  F& |" `M;
. @2 W9 q; v$ v8 C8 @/ j$ yNumberField(M);
* Q( V7 A/ B4 g/ u# r+ h0 D6 L  hS1:=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;
) Z) _* M$ J2 C+ h  d) aIsQuadratic(Q5);
. W4 w* N. i) A# IIsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
5 E: a5 t& c& }2 T) S3 nIsQuadratic(S625888888);
Factorization(w^2-50);  
6 {( A7 e7 f3 l" M* r  p/ sDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
& R- Y! p" |8 H. UFundamentalUnit(M);
5 T) R. f$ C' pConductor(Q5) ;

/ M. u3 S. {) g3 M0 rName(M, 50);
4 a* \& C0 d, fConductor(M);
ClassGroup(Q5) ;
1 I4 l" f5 q0 V) h" H1 p8 _ClassGroup(M);
" F8 Q5 Y, ]  p$ F! pClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
$ \# n) T& ^, Z* q7 ^
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
0 k2 V' ?. ~' _  q6 S1 [NormEquation(Q5, 50) ;
NormEquation(M, 50) ;

Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
2 L7 n" `8 }$ K+ V7 QQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Order of conductor 625888888 in Q5
0 c9 W2 U3 c! @7 Ftrue Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
3 @/ V/ {0 K) p; b" R7 }( ntrue Maximal Equation Order of Q5
( F$ ?  d# w1 gtrue Order of conductor 1 in Q5
1 R4 @  }5 @  P. ztrue Order of conductor 1 in Q5
1 }+ C& V1 W" P2 Z6 v* H- \true Order of conductor 1 in Q5
2 t. p' `$ O: K* L" j, W[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
& `4 c6 b9 \1 b]
* l/ Y4 w  W3 _1 Q1 q) P' w8
Q5.1 + 1
- a% e! K1 T4 s3 }$ l# B$.2 + 1
  ~0 f8 l1 S" \/ y) t9 z8
7 ?* D& @9 Q) n; i4 Y) ~, t+ ?
+ U! J# v) z  W3 O9 w7 h>> Name(M, 50);
       ^
) ?3 [& `  X. u' x, l  M) P  CRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
0 ~8 K1 _  T' ~! k
9 i4 m, Y; a: ]$ I1 D* ^1
) Z) [" \" r6 a3 a' c! QAbelian Group of order 1
, n* Y8 T. V$ G7 C  g# {* PMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
# N  R  Z, d' W8 ZMapping from: Abelian Group of order 1 to Set of ideals of M
! [8 D+ j' M7 A1
1
Abelian Group of order 1
, w& u/ E9 v% r: {1 C: w" [Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
5 @2 V% O# b3 D; q, a' TAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
9 L+ W  X+ U0 r. G6 [Abelian Group of order 1
, y: u# b% }+ c, j" k& [) zMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
1 @/ |# x7 F. H  n1 A- a8 j0 l( `8 given by a rule [no inverse]
true [ 5*Q5.1 + 10 ]
3 H" x" ^- [$ D" A" G/ ?true [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

; p' ^  m  A, b: f
基本单位计算fundamentalunit ：
5 mod4 =1                                              50 mod 4=2
: n" T1 f2 }. l) A/ E+ t7 c  X
# G, q; E& a4 C4 h x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
2 q- Q, a: e! g) p2 M% C" i0 k% b x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.


最小整解（±2,±1）                              最小整解（±7,±1）
                                                             ±7 MOD2=1
% J: D9 @7 \3 u6 [
* m4 a/ E! @: s7 @/ @5 ?两个基本单位：

lilianjie1

lilianjie 发表于 2012-1-4 18:31
" `6 f# L1 R& m9 s基本单位fundamentalunit ：
5 mod4 =1                              50 mod 4=2

基本单位fundamentalunit

lilianjie1

8 A9 K' ?3 }& r) e判别式计算Discriminant
1 G5 X8 Q: U  X( H& A% B
# _- Y% Q3 Z$ W  |. ]% t5MOD 4=1

0 a7 L8 ~7 U7 E（1+1）/2=1          （1-1）/2=0

$ ~& z5 ]1 g( l7 Y. _D=5


50MOD 4=2
D=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44
4 R% T# L. I4 p+ x
分圆多项式总是原多项式因子：
C:=CyclotomicField(5);C;
  S9 S1 z3 u$ l; ECyclotomicPolynomial(5);

分圆域：
! k5 I8 H1 j9 F# c. {: T- J分圆域：123
/ U' m+ ]6 P) y# Z. v) t
, q) ^0 k3 b) i0 ?R.<x> = Q[]
F8 = factor(x^8 - 1)
  {& n' k. A+ I- @& @$ tF8
) R: @: Q& y& I2 n# |# @# ?3 Q' v' z+ B
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
$ T: O& o" W, |+ b# {/ Y5 r. L
5 Y$ t5 d- p" B4 X4 q4 z  T1 ]Q<x> := QuadraticField(8);Q;
- J9 V& \. h2 f0 WC:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;

% U$ l' E7 k4 N7 Y8 z& PF := QuadraticField(8);
9 M6 L. ^" c7 C2 k+ {; b3 a$ O% J; j4 nF;
D:=Factorization(FF) ;D;
/ [) v+ z$ j2 z5 B1 f+ W# ~Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
# C/ ]' A1 V2 g1 O$ sCyclotomic Field of order 8 and degree 4
$.1^4 + 1
% H) i. {0 d  f' ^: yQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
; z$ T( a  P) u; J0 h[
+ b' C- j% ?8 P$ A% q% g- `    <$.1^4 + 1, 1>
6 S2 G0 S# A3 T+ I# i]

R.<x> = QQ[]
F6 = factor(x^6 - 1)
# h5 ~& |3 k+ S' l2 kF6
9 t# [7 z  p" g1 \7 A2 G1 D- T
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
7 p: G1 E! O) G. A) Y# F  |9 g
$ x7 M5 ?: `/ z) g( X) U* uQ<x> := QuadraticField(6);Q;
  K5 K9 V5 K1 A( M; lC:=CyclotomicField(6);C;
( q& b$ n5 m7 K' V: w9 bFF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
F;
D:=Factorization(FF) ;D;
( B! M" g# {8 ^4 c* \Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
& p7 V4 {3 G  e( ~7 o& X" ACyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
' W: J) ~# A9 T$ z2 u8 S7 U/ VQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
- c5 T5 y; Y8 Q3 ]# l[
" ]1 Q# @/ a# O: q0 q8 z    <$.1^2 - $.1 + 1, 1>
5 v+ \4 }5 b0 x# f+ I0 @; T* A]

1 X8 O3 ~4 e5 r: A$ @R.<x> = QQ[]
1 c9 ~3 @3 p. q9 g% w; o3 X1 v) A; YF5 = factor(x^10 - 1)
/ p( s/ O2 r! [7 h2 O0 aF5
2 L( J3 s% f8 N(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
* q( }% O5 w5 [' c$ u- a) j1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
5 T" i' g2 T4 A; X9 z
Q<x> := QuadraticField(10);Q;
" v+ l* i* L$ K" KC:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;
( t! e( I7 R  F4 ]( G
F := QuadraticField(10);
6 n; v- Z' C0 K- qF;
" q' M2 @# ^: v) P0 g$ cD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
* x' {) R, `' G1 W$ P3 M$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
$ Q1 ]4 t; A+ w# n5 b! F    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]