实二次域(5/50)例2

lilianjie

1 L( \5 s; @& \: S% A$ x0 a+ H; W3 J
* U0 q2 i( T' R- MQ5:=QuadraticField(5) ;
Q5;
Q<w> :=PolynomialRing(Q5);Q;

EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
NumberField(M);
+ p& f' e- P# o' qS1:=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);
5 ]3 T0 M& T. ~9 WFactorization(w^2-3);
Discriminant(Q5) ;
4 }- \1 z" n; e$ I- ^4 UFundamentalUnit(Q5) ;
FundamentalUnit(M);
- X3 S; z* X9 z! k1 uConductor(Q5) ;
Name(Q5, 1);
Name(M, 1);
2 C0 N8 K2 Z& V( H1 v( g7 N' ]Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
5 E8 R+ a; s% e+ Y1 EClassNumber(Q5) ;
% F% Q4 e  C+ Q; kClassNumber(M) ;
* k8 }6 B# W4 U$ i
PicardGroup(M) ;
3 @  Y) n/ g  M, n& y2 m4 sPicardNumber(M) ;

8 b7 F- \1 T- a' x
QuadraticClassGroupTwoPart(Q5);
2 I# ^2 z% y1 g8 j/ ]1 x# [4 IQuadraticClassGroupTwoPart(M);
+ \  E: x7 ?1 c1 w+ c) \, i
. R0 Q% c! Z* `6 h  A/ X2 m; u/ x
' I' i! g, q3 @: U0 oNormEquation(Q5, 5) ;
NormEquation(M, 5) ;

0 p! t! o8 M0 L# @4 I/ e( R
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
% b4 Z; L5 ]3 TUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
" P5 Y2 S- t' i# I7 SMaximal Order of Q5
2 a+ F! E2 f8 y1 w- {# h" g. K+ f; lQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
true Order of conductor 16 in Q5
true Order of conductor 625 in Q5
1 U7 _& \6 ?  [" p- |, {, Ktrue Order of conductor 391736900121876544 in Q5
; ?2 W# R( {/ Q3 C; U( E) ][
& c6 _$ W2 G, O- W; h9 T4 E    <w^2 - 3, 1>
3 L* M. E! z8 Z]
5
1/2*(-Q5.1 + 1)
- x; f8 z7 ~& ^; j# H-$.2 + 1
0 U+ f4 R, T  Q. f0 |9 a$ Q6 S+ @5
Q5.1
$.2
1
9 Y5 I8 @' L  t- G9 `Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
: u. {( d- j, N+ a/ y% nAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
4 E6 C: N6 k; ]8 K4 u- }1
8 n" q( d. X, i5 |0 E5 H* q" D1
Abelian Group of order 1
# ^- J0 A- s& x( M3 K/ N; K% ZMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
; O- E7 D: j/ D: Tinverse]
$ m9 d+ v: t1 B' ^3 j  k! d1
" ~0 S) h, [, u3 a" ~: TAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
! y6 f* ~  |. m: I" ?5 given by a rule [no inverse]
8 d" V6 O8 U) c! PAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
& o' c$ u* m; S; f" u' k; btrue [ 1/2*(Q5.1 + 5) ]
true [ -2*$.2 + 1 ]

& m8 c; ?& \  J: @4 L4 w
+ \( w0 P. f- a2 x1 R& }5 T

0 U+ e0 Q2 {6 c$ _, K

0 w0 P& z) }8 r4 x* Y4 \

( ?! o6 M& f4 x

7 t) s" y9 t8 b8 O; _& r' g+ S) V
==============
! c, O$ c( X% r2 x
Q5:=QuadraticField(50) ;
& J" F6 h: d! ?! wQ5;

# t! F- l- j1 ?/ [. v6 lQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
9 R0 P. V1 w4 O& W) \$ yM:=MaximalOrder(Q5) ;
M;
# k( g3 N8 ^$ W$ c) W8 YNumberField(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 Q& W' E4 o* _& m1 w$ {: _+ jIsQuadratic(Q5);
+ n1 ?" {# I7 z; s* x5 I- Q( qIsQuadratic(S1);
# R+ K. H! Z& d3 KIsQuadratic(S4);
IsQuadratic(S25);
+ W4 O6 Z6 l# m- ?: E) @' DIsQuadratic(S625888888);
Factorization(w^2-50);  
, S1 P/ ?6 ~7 G, G: WDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
% t3 X$ A. \+ g
3 c; u; l" N4 x  x) Y- Y0 TName(M, 50);
% q/ `# Y+ B; q6 {4 K' A% l' eConductor(M);
; Z8 i/ v9 Q* p4 i% oClassGroup(Q5) ;
ClassGroup(M);
9 Q& \% D. Z4 k3 _$ c: ]( aClassNumber(Q5) ;
ClassNumber(M) ;
- \, J9 F" F- B+ I. fPicardGroup(M) ;
- t- _" n5 w7 Y$ i; ZPicardNumber(M) ;
0 d# r6 `- I' P' B2 f9 w
QuadraticClassGroupTwoPart(Q5);
) Q. y$ {9 z8 r  RQuadraticClassGroupTwoPart(M);
$ O2 I- J# @6 g' _' ZNormEquation(Q5, 50) ;
NormEquation(M, 50) ;

Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
1 p9 y' m$ \% a9 d" M+ _. k& H+ `Equation Order of conductor 1 in Q5
8 p: G( d- d- [* F+ U5 D! ZMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
6 i/ }9 Y% e( KOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
$ X# Q1 @- r. {true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
5 i- i' Z0 s6 o9 |5 v: Gtrue Order of conductor 1 in Q5
[
    <w - 5*Q5.1, 1>,
& G2 Z0 \( \$ V- Y0 t    <w + 5*Q5.1, 1>
! |, k0 E, ]7 h8 B* t]
- m1 q7 E$ w% }; W8
# _+ n& X# x7 z0 Z" ~: YQ5.1 + 1
$.2 + 1
0 A) ^1 a8 |' Q; e* _& b8

. H& C( y8 q. L2 w>> Name(M, 50);
* k& s# Q) [7 |$ T       ^
Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
4 ^3 F! n2 B; M
  F2 ]) B# f7 l$ t( q. |! _1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
; l( _; U/ Z4 H& p# `Mapping from: Abelian Group of order 1 to Set of ideals of M
1
2 `$ N+ I1 U, A# p6 E5 ?$ @  }- \1
0 j0 y! m" N+ M6 _3 q; t' XAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
3 Y* Q: G1 H% Z0 ^: @) F, y& M1
Abelian Group of order 1
4 B! R$ z) J  r4 ?& W2 v" \Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
( X3 z, N- d' u8 given by a rule [no inverse]
: o1 l3 g& z* d' W( f0 ?4 z: {Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
" v! L1 f7 Y: T/ q9 Y8 given by a rule [no inverse]
true [ 5*Q5.1 + 10 ]
  @$ @2 q! N2 [8 H# D1 c# H/ F$ gtrue [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

" d3 F9 _6 v0 O5 |  {) l
基本单位计算fundamentalunit ：
7 J3 V4 |" q- z  s4 U4 z$ W5 mod4 =1                                              50 mod 4=2

9 Y1 x; E( ^0 A+ S8 x/ E3 N x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
5 N& \& m) @! e' A x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.
1 m$ o) G  R3 l; V) O
0 ]6 u. f* d! i7 I" m
/ V( P8 d7 ~# S3 o最小整解（±2,±1）                              最小整解（±7,±1）
                                                             ±7 MOD2=1

! l( k: |( \  u  V% j两个基本单位：

lilianjie1

lilianjie 发表于 2012-1-4 18:31
! l* t7 w1 H" ^4 g% {基本单位fundamentalunit ：
5 mod4 =1                              50 mod 4=2

6 _9 R6 E' N/ ^6 Q基本单位fundamentalunit

lilianjie1

4 o( x3 M) ~) z) Y5 S+ L判别式计算Discriminant
* T( {& }2 D: s. E) ~; ~
5MOD 4=1

* ]# G: y0 ~5 s) p" j, i: q; z（1+1）/2=1          （1-1）/2=0
$ O. B* g- j6 ^! A1 ~, A' G, v' O
D=5


$ {/ n" z7 b8 c' i! p! J50MOD 4=2
D=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44

/ d5 m7 S  y6 `, ^- Z1 f分圆多项式总是原多项式因子：
C:=CyclotomicField(5);C;
. g2 C3 B0 M0 Q2 j) P0 O' `5 tCyclotomicPolynomial(5);

分圆域：
( ?3 ~. B  Q2 U% \' d3 ]# i分圆域：123
% k7 {4 [4 T: D3 u
2 A5 m/ \: E. I5 Y4 j; jR.<x> = Q[]
F8 = factor(x^8 - 1)
F8
+ h" B) {  E6 F- y
9 M% B: d: W% j8 W& l9 ](x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
2 t. d/ L6 c4 N
Q<x> := QuadraticField(8);Q;
+ ^/ N. i/ S, `& GC:=CyclotomicField(8);C;
" \4 _7 U4 h8 C' h% b! f5 Q1 w6 w2 S% mFF:=CyclotomicPolynomial(8);FF;
, k. G- R$ \/ Q
F := QuadraticField(8);
2 s& u6 q  Z/ p( w3 m& JF;
  G& I* |3 \) J: K! Q" kD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
$.1^4 + 1
1 l( z0 b+ N% j& r1 F- k; ~, kQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
) v1 A/ ^  J* p1 }! |, b) A[
* s1 a$ s. N2 b  T    <$.1^4 + 1, 1>
]

- t- _- Y3 H% }: c5 d! ?& E% g! mR.<x> = QQ[]
0 v9 \/ N$ d0 i( F2 F$ LF6 = factor(x^6 - 1)
F6
& v" l9 q0 N) P8 u( ]7 K8 Y( [
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
! `8 ^% N! @* _1 y* ^# Y3 M
0 T/ z/ h  G6 f/ jQ<x> := QuadraticField(6);Q;
- k5 W  E6 u. v, x. PC:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
( ]$ J; a. x2 g2 ~0 ZF;
D:=Factorization(FF) ;D;
6 x9 A' v  Z7 U7 y! @, AQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
) |0 c( X. L- ?% [, h$.1^2 - $.1 + 1
2 \8 C  e5 o! O( I/ @( `Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
  E' x  @" g) n3 Y6 O) }- P    <$.1^2 - $.1 + 1, 1>
]
+ k1 p& u5 C7 n) A2 M
3 d3 Z* q  d2 s. d) C  N: `R.<x> = QQ[]
F5 = factor(x^10 - 1)
' m, h8 T+ O& u" \- fF5
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
9 M* k+ d7 O. c& r  i3 p1)(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);
+ h5 _4 x+ A1 [5 D$ g- ?" rF;
D:=Factorization(FF) ;D;
5 K8 B% n3 z+ c! |. Y( @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
9 @; x* x2 c( p/ V0 [& VQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
% L( u5 u# i9 \% U' C' }! N& x[
; q. J2 @& u, e& i; \    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
  ]. F( c% |5 x" a]