实二次域(5/50)例2

lilianjie

Q5:=QuadraticField(5) ;
Q5;
Q<w> :=PolynomialRing(Q5);Q;
' c+ W( O' z# S) t6 G
EquationOrder(Q5);
- m9 a; W9 a, ~M:=MaximalOrder(Q5) ;
M;
NumberField(M);
/ N2 _( `6 K/ R- B2 z! \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;
! s8 i8 c  F# FIsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);
Factorization(w^2-3);
( S0 W: D7 u- v+ NDiscriminant(Q5) ;
, C) a# D% r6 s: G2 P! TFundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
* i( ~  B. ^, u3 U) FName(Q5, 1);
: \9 Y% k0 |! \' r/ s2 Q( ?9 D% s6 G' nName(M, 1);
% v3 l* e4 E8 S. P' n; Z$ BConductor(M);
ClassGroup(Q5) ;
( W; ?4 Y7 V1 r1 A1 VClassGroup(M);
ClassNumber(Q5) ;
; K/ D" L: `% S2 N! z7 a7 `ClassNumber(M) ;
  }5 _0 V( D( L% ?- U4 ]+ z. O$ f
  K" C1 P7 h. G3 T# y1 T  J4 H" N  vPicardGroup(M) ;
3 r' X. B0 O' `& c  M& Z" {4 l2 f& {PicardNumber(M) ;


+ e4 C; {- Q- D6 X* KQuadraticClassGroupTwoPart(Q5);
: o* F: s4 f0 E; rQuadraticClassGroupTwoPart(M);

0 ~, p; r% ?) b5 Z2 \4 Y) \6 @
NormEquation(Q5, 5) ;
NormEquation(M, 5) ;
! L! I' C$ |/ M. j$ E8 k. Q

4 i. a, q# D* G5 g9 {, c4 O2 H) OQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
. t1 V/ y$ D* _$ oEquation Order of conductor 2 in Q5
Maximal Order of Q5
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Order of conductor 625888888 in Q5
( g# W6 A% y" R7 @true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
true Order of conductor 16 in Q5
8 ^7 Y0 v  B1 _( b5 J' c1 [! Atrue Order of conductor 625 in Q5
' u& e" x1 c+ h, ?& X# Xtrue Order of conductor 391736900121876544 in Q5
[
    <w^2 - 3, 1>
]
5
0 a. ~; k9 E7 E1/2*(-Q5.1 + 1)
-$.2 + 1
* P' B. s/ d7 c7 k1 ?: `( S( ?5
& w4 ?5 }$ H- ^6 z  dQ5.1
) `6 g6 D& M& j, I/ V. f$.2
) \  x  N1 T& p9 {# J" ^1
1 ~7 I6 Q' R) PAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
5 o, u+ L7 f( `, r! C: X1
1
Abelian Group of order 1
* a$ F/ x3 u8 E; d3 G" fMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
; T/ [# i. d5 @. {7 o3 n3 mAbelian Group of order 1
3 N: F5 y: L" [0 ^Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
6 b1 b( b: ^- v( m# E( J+ S/ QAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
% {1 v  A& D$ h0 `5 given by a rule [no inverse]
1 `. k$ z: y7 j9 Y3 g9 _* o; `true [ 1/2*(Q5.1 + 5) ]
true [ -2*$.2 + 1 ]
: V9 L" s# J! f# g! ]
0 J, v" f3 G- A8 D2 T


) d  o- p% y* l( F8 ~" Q' k
0 c! p8 P% W7 T* E

& [/ g" d9 b* ~1 F( m* a+ K7 H3 z


5 `  o9 a5 P( h
==============
$ L9 A3 ~+ M5 ]3 ]
1 [6 T- _4 i/ P, w! y0 i. U, c; TQ5:=QuadraticField(50) ;
Q5;

% M  v( x8 z+ f% y+ s) oQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
( m! M+ [% ]2 [* N5 B8 C! rM;
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;
6 G: x; [, @$ zIsQuadratic(Q5);
1 b, g+ {+ G1 h* p0 J0 C7 CIsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
% |1 U6 i) v8 c: n& D! vFactorization(w^2-50);  
% p+ z9 A' o+ ]7 P9 g9 tDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

; x* G9 Q2 l; N( XName(M, 50);
; w% n! j. n# Y: f5 \4 v+ V& g. T' {Conductor(M);
) ~) Y) ?  [, n( @ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
$ t7 n/ a+ P! f; P/ tPicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
$ r" A* Q6 M% ^4 @0 l1 QQuadraticClassGroupTwoPart(M);
7 G( {  g/ o. Q. N6 l! X2 kNormEquation(Q5, 50) ;
NormEquation(M, 50) ;
1 Y7 i2 `! w6 N4 z$ d7 [6 u
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
+ u. b( B  _& c$ M& yQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
# A& D' @) f/ n- aOrder of conductor 625888888 in Q5
7 {9 F3 [9 t2 y' qtrue Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
true Maximal Equation Order of Q5
/ K" _3 r0 j  {3 A3 Dtrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
7 D  g( ]8 R4 p' ~true Order of conductor 1 in Q5
[
    <w - 5*Q5.1, 1>,
2 L8 \  x4 z7 U+ J    <w + 5*Q5.1, 1>
]
6 ~, Q1 a3 b' i( T! X  L8
Q5.1 + 1
3 w0 {0 K  e. K) ~$ q+ j$.2 + 1
8
  L( S' c+ b( [9 ~
" i+ f6 g: v: l>> Name(M, 50);
       ^
7 a$ V; F: t7 M- J7 R  H1 W$ BRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
$ s" w* x2 q) j' P! g
& P! y8 p1 Z6 d( l1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
0 ?' S! \. H5 K( d. m, R/ ~: Q1
1 b4 t2 U/ _( y0 F( k2 g8 NAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
8 x1 g5 h% h  iinverse]
7 o% ~3 R" C1 r. C, A1 F$ ?1
) r* A$ t3 B" a/ M' 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]
& V( |" S. y3 q# y( jAbelian Group of order 1
- E& O% x( {8 l( t. bMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
+ E3 b% |! R5 w3 ]8 given by a rule [no inverse]
true [ 5*Q5.1 + 10 ]
0 s, v! m( J3 {/ q, Rtrue [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

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

9 s1 G# H+ j# s3 h x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.
) E" }. B7 t9 j2 T
7 \5 z, C; A1 H
- r; Z3 d8 y! j( g9 h最小整解（±2,±1）                              最小整解（±7,±1）
9 u; @  j1 \( o: d& \8 c                                                             ±7 MOD2=1

两个基本单位：

lilianjie1

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

基本单位fundamentalunit

lilianjie1

判别式计算Discriminant

1 p- I. l4 ?6 K8 d5MOD 4=1
- b- }& s6 L5 Y8 ~
（1+1）/2=1          （1-1）/2=0

D=5


50MOD 4=2
, i; Z+ y! W& F1 n6 |0 P; W$ o% W: A! QD=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44

$ ~/ v8 x3 Y! h: x( L+ |分圆多项式总是原多项式因子：
8 u% V" g8 L/ E% V/ s- W& ^C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

. J1 f5 E! G3 {( v2 Y. u; d9 S
! |9 d- ^# T" H分圆域：
5 H4 ~6 x3 d, E2 `2 F! e5 i+ o分圆域：123

7 A0 t' Q% _  Z; {- N7 H8 IR.<x> = Q[]
+ c- K, F, d) }F8 = factor(x^8 - 1)
7 P3 w1 v* x1 YF8
5 Y/ a8 m- {* L% ~1 I
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
6 I/ z( @4 z. K+ Q9 L8 K% n) w
Q<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;
  l: W) R7 m% L1 l
5 K; J; Q9 q7 _# I7 rF := QuadraticField(8);
F;
' Q7 f5 P; H4 z( z, `; CD:=Factorization(FF) ;D;
( P% q& _4 l" h7 PQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
( ^. d- Y! o: R* U+ ]3 @. `: K' dCyclotomic Field of order 8 and degree 4
$.1^4 + 1
+ ?: l( C3 r. NQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
# L# I/ `7 ^$ G[
    <$.1^4 + 1, 1>
; G9 Y% e* O3 G3 K]
: C$ ^0 r0 e6 ^: X: i
R.<x> = QQ[]
. M8 [" }$ |& G) ?: N& zF6 = factor(x^6 - 1)
8 c/ k/ W: l. hF6
' Q: G( A' `, E, m5 P+ b  w. V
. ~" ?3 r! e) s8 A* G- |. y5 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)
) y. \6 R* E8 P) {! z% m
Q<x> := QuadraticField(6);Q;
4 o- ]& i# t6 n/ kC:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
F;
0 g& p- Y, E7 j9 A3 OD:=Factorization(FF) ;D;
; X; \7 o; `2 X* Y1 J$ kQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
2 t8 W0 s. @) `0 f1 s4 d2 Z+ [$.1^2 - $.1 + 1
  y5 w; }" ]! v% `Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
, L0 c/ @3 C8 b- h[
9 x6 J' \6 t: M4 e    <$.1^2 - $.1 + 1, 1>
# J/ N9 F* J; j3 d1 y. G]

+ s1 I" w' J- Q- iR.<x> = QQ[]
F5 = factor(x^10 - 1)
; f* f7 Q) v" i: Y$ E: F1 uF5
9 E& R3 N7 t" @9 L0 t* l(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
5 z( D. g# Q& H1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
& l/ P! \- m( q2 j- A' h8 I; j% r
Q<x> := QuadraticField(10);Q;
* V4 q8 z  O, b4 F5 p+ u2 lC:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

; ]0 _( S7 F  F: g0 S5 HF := QuadraticField(10);
F;
* R5 V% @  j$ q% d1 fD:=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
1 Z; @9 U; @# T. G2 ]Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
4 A% r+ I  v0 }4 c9 _' @4 c2 Y    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]