实二次域(5/50)例2

lilianjie

4 o# d! H+ V! J! R5 dQ5:=QuadraticField(5) ;
; B0 k! {$ I" O5 WQ5;
Q<w> :=PolynomialRing(Q5);Q;
' Q- ]- |* X4 e3 e' ?  L2 S  {
EquationOrder(Q5);
) ~0 {$ I6 N- w/ o6 X6 W  Z+ nM:=MaximalOrder(Q5) ;
0 k( e9 {( q- B( r& x4 z3 jM;
NumberField(M);
- q8 Q3 J$ x+ C! ~! wS1:=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);
/ X5 ]9 @( Z% k" zFactorization(w^2-3);
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
+ \" X2 H+ S! }9 uFundamentalUnit(M);
Conductor(Q5) ;
  D- ?" {+ R# N; E9 p# K4 CName(Q5, 1);
& i/ a* l# i& m2 Z/ K( vName(M, 1);
% R' R! S* y* H( H& z( I) M* zConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
; _: A3 ]' \5 A1 R  X( X$ Z4 ?) vClassNumber(Q5) ;
ClassNumber(M) ;

, V0 I4 j+ {1 P# O8 }4 IPicardGroup(M) ;
8 N+ v7 e& t+ iPicardNumber(M) ;
# _+ P1 l6 d4 a( A/ r% d
& I" E% J% h% a
QuadraticClassGroupTwoPart(Q5);
* j# U3 w1 W* x1 k3 @QuadraticClassGroupTwoPart(M);
/ O: a$ c4 l/ O8 K
" }/ l: x7 d$ b# v) i9 V
NormEquation(Q5, 5) ;
. p  b! V  n) F$ F! oNormEquation(M, 5) ;
  g4 k$ z  I; R; U3 T- b

Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
* q2 l4 g  g: w4 YUnivariate Polynomial Ring in w over Q5
* z7 U. v& D" }! X  ZEquation Order of conductor 2 in Q5
Maximal Order of Q5
7 t/ F! }4 r: ~) Y; y$ b  SQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
. l; s; u6 i1 O' e% bOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
8 ]; X! n# z: [/ A. B0 itrue Order of conductor 16 in Q5
true Order of conductor 625 in Q5
* ~/ A9 O5 s0 ^- D+ v4 k, A1 E! vtrue Order of conductor 391736900121876544 in Q5
[
    <w^2 - 3, 1>
0 y* n# P. O  e( Y2 J3 P- S]
5
, n3 [% Q1 O4 ]  r  Q$ c4 |, x1/2*(-Q5.1 + 1)
-$.2 + 1
2 h) x, c6 \/ @7 u1 X9 i( ^5
Q5.1
5 c6 t- l0 I" Q4 B6 E1 ~. [$.2
' s) ?5 J: w8 I1 i' T6 [1
Abelian Group of order 1
* h1 a1 q% F7 r# _+ I% S: y% L1 o3 AMapping 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
  L( Y7 f8 o, }2 R1
1
, B6 [) @+ l: e/ `Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
3 a6 b1 n+ G7 o/ b1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
" i% b" I0 O( i2 m; o9 O0 l5 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
true [ 1/2*(Q5.1 + 5) ]
& p/ K* R5 A+ E' t( gtrue [ -2*$.2 + 1 ]
$ y5 I9 @0 J& O" x
! G2 w/ k1 S- H& u
! @+ ~' E0 G9 {9 p
" ?& g8 C6 F1 x7 f


6 J" v! ?7 H0 x& W




6 n! o0 D, e7 U% H2 }$ U* P5 H7 j7 F==============
+ Q3 v, ?8 H1 y, d/ P5 Y0 R
8 x2 ]  y# b5 C- n9 q1 x0 y, `+ }Q5:=QuadraticField(50) ;
Q5;

6 y' L3 i9 {  E/ X2 k: zQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
% M) ?# a# t% @8 QM:=MaximalOrder(Q5) ;
M;
# z# M+ e" w; B: g/ _. XNumberField(M);
/ ~  j) u) D3 f  YS1:=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);
- p7 H1 ], j$ |' [IsQuadratic(S1);
IsQuadratic(S4);
" e' K" s/ I9 J$ \! o9 MIsQuadratic(S25);
, e: E# u! i1 _6 B* v  @$ V+ mIsQuadratic(S625888888);
Factorization(w^2-50);  
Discriminant(Q5) ;
2 a+ r' {, b! E5 |FundamentalUnit(Q5) ;
1 Z# z0 a  {# ]8 r3 s" [( ]* FFundamentalUnit(M);
Conductor(Q5) ;
: k1 {& E- y, |! ]
Name(M, 50);
Conductor(M);
ClassGroup(Q5) ;
2 a# m3 j8 L$ l8 B: O1 KClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
( U" A! |9 \+ [! w5 G& _PicardNumber(M) ;
5 Z8 s* g3 @7 V: G
QuadraticClassGroupTwoPart(Q5);
! F! }2 O/ _$ d( YQuadraticClassGroupTwoPart(M);
0 \$ C/ j6 T* T, d6 W% C4 h+ bNormEquation(Q5, 50) ;
NormEquation(M, 50) ;
& s% A" w; R; B2 U" _3 V
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
( e' ^7 {1 _1 U3 T/ v! _! s& V% KUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
' j' v2 w% }- R( g6 C0 CMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
1 z, E) ]8 g5 j8 D! T; J5 u0 LOrder of conductor 625888888 in Q5
; P: C- n2 v! L+ Qtrue Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
& V3 v3 H" _; E: I1 Itrue Maximal Equation Order of Q5
6 |+ s! z& {" g: Y4 N1 B2 I+ }true Order of conductor 1 in Q5
" e* x5 H, v$ X" L1 _  strue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
2 }9 W& T6 ]0 U) Q* V5 f[
3 V. V* U4 G$ R3 C1 H    <w - 5*Q5.1, 1>,
* W7 l  y! \$ f6 Y2 g3 [    <w + 5*Q5.1, 1>
]
8 ^( K- [/ D' W* P8
Q5.1 + 1
/ F! H; B* c% l2 a  {; I/ e$.2 + 1
8
1 A% c( ~1 Q; q* X# u
>> Name(M, 50);
       ^
Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
! i8 w' I, [6 u; a) w- b9 D( l
1
Abelian Group of order 1
3 @& b0 P: V- v$ l% WMapping from: Abelian Group of order 1 to Set of ideals of M
: x& {% J* C$ }! M/ u4 n1 vAbelian Group of order 1
9 _$ Z- k" d+ H3 ]Mapping from: Abelian Group of order 1 to Set of ideals of M
1
1
Abelian Group of order 1
8 }* t6 x# L  g! ~" dMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
- h% y& _' J0 ^' ^0 |5 F( B- ]inverse]
1
& f, C; y' W- A  x$ `+ sAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
; ^4 _: g$ G- f6 Y% a8 given by a rule [no inverse]
Abelian Group of order 1
+ s8 j6 ~1 n  P4 ?& [( ~+ ~5 MMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
( Q+ S, E, O$ w! X; I0 V3 C0 rtrue [ 5*Q5.1 + 10 ]
true [ -5*$.2 ]

lilianjie

lilianjie 发表于 2012-1-9 20:44
% M. i) `; o3 Z! D+ q7 O' y( J) `
分圆多项式总是原多项式因子：
4 f- ]1 W  L/ o- P" @C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

% {% G4 S7 v8 f7 `; ^2 H分圆域：
分圆域：123

! c4 j" U. n/ [! o. rR.<x> = Q[]
3 {1 x% h0 e( g/ w  ?4 r' H' v7 jF8 = factor(x^8 - 1)
F8

(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
' ?4 J' c! z( ?3 c2 b; P
* B" x6 r3 a1 _Q<x> := QuadraticField(8);Q;
6 e1 R/ e6 n% J, e7 s! m1 g2 k- AC:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;
, x) W) @7 m# W
F := QuadraticField(8);
; c" S2 p& i9 P5 I4 LF;
! e) m- F, G) u1 T$ WD:=Factorization(FF) ;D;
$ F1 k+ n5 e% @4 CQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
4 O" e% [( e+ hCyclotomic Field of order 8 and degree 4
$.1^4 + 1
* U1 j* F- f/ L7 aQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
    <$.1^4 + 1, 1>
]

+ i/ @5 _. n- K) s7 s! s+ H( W( ?* p2 lR.<x> = QQ[]
F6 = factor(x^6 - 1)
# u* M  Z% ^7 l, r- l2 |3 F$ O: P" ZF6
0 V: @% j7 Q: d* |- l1 v
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

  X1 {8 {7 V3 [; V" FQ<x> := QuadraticField(6);Q;
& d& r9 l. J+ f0 vC:=CyclotomicField(6);C;
. B; C3 @! l# U8 o8 jFF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
* \/ u. r/ Q& T; U6 H0 vF;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
- I, j! W* _! H3 f4 s. `7 d6 aCyclotomic Field of order 6 and degree 2
5 A! i2 o  [8 S5 U  `) {$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
' e# b1 k0 S) ~) U# Z    <$.1^2 - $.1 + 1, 1>
+ V: T0 f, {0 ]! O" W]

R.<x> = QQ[]
9 {' Z0 P! u- Q$ A  xF5 = factor(x^10 - 1)
F5
6 N' V9 A2 F" [% P& E(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
# {2 Y& f6 P: m, @$ u1)(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;
7 b+ @0 C6 b2 B# D( H3 Y. w: \C:=CyclotomicField(10);C;
  a1 \) H+ A5 I" i  UFF:=CyclotomicPolynomial(10);FF;
5 b% ]+ n: M1 i" {7 [+ G: y4 {
F := QuadraticField(10);
, l1 P0 E' U9 v2 w. X% E8 zF;
8 j) C6 p- z, }* r5 E7 U) K0 ]: sD:=Factorization(FF) ;D;
) W+ `$ F* k& C+ r- @Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
! ~/ J9 h. ]/ O" s' B$.1^4 - $.1^3 + $.1^2 - $.1 + 1
% I9 Z% x- @+ j- J5 w- F- NQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
8 B- P* X5 W# A' B& t7 P[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

孤寂冷逍遥

lilianjie1

: c- O6 \4 b8 {: a3 p4 Z5 `* f
0 I  Z( v4 G' I1 i. z, J判别式计算Discriminant
3 z$ g$ d, ~. ^; s$ B. m  R$ L# H. w
5MOD 4=1
# c/ W. b7 _) k
6 o+ @* D2 c5 e1 i8 g8 B（1+1）/2=1          （1-1）/2=0

D=5
5 I$ p+ a" C5 k

50MOD 4=2
) F/ k+ {( s$ ~$ @3 vD=2*4=8

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

lilianjie

. f0 k  P) N. r+ P+ S& n; |# c/ Z
基本单位计算fundamentalunit ：
5 mod4 =1                                              50 mod 4=2
! _, S6 o2 \) B' B+ L& e7 W
x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
4 [/ f' ?  h+ M- }# s+ } x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.

1 c6 \) |( U' K* _+ j( h
/ W; Z  u4 T" a7 |最小整解（±2,±1）                              最小整解（±7,±1）
                                                             ±7 MOD2=1
; x! V8 P  [+ p* X. J8 R
两个基本单位：

lilianjie

二次域上的分歧理论