实二次域(5/50)例2

lilianjie

: b, E$ |: @, \, k2 @' k
Q5:=QuadraticField(5) ;
Q5;
2 r9 x* H- }8 ~# S/ ?* x+ W% v7 IQ<w> :=PolynomialRing(Q5);Q;
8 B% l* K3 A, Q- @
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
8 U+ e" d* {5 Z, l# j- r8 D$ FM;
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;
2 b7 M$ f9 B, f; l. |3 J. ^IsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);
6 Z6 J8 C# @/ }Factorization(w^2-3);
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
3 p/ m( n$ f1 M, AFundamentalUnit(M);
: T8 d7 o+ p2 j9 n+ B6 n0 S0 T3 IConductor(Q5) ;
( F/ E. B  V5 U7 S# X$ c9 yName(Q5, 1);
2 U  v, G; W7 n9 w) I( eName(M, 1);
Conductor(M);
  o5 j6 `7 @5 x2 o# M7 t, ]4 XClassGroup(Q5) ;
ClassGroup(M);
: c" b$ {' [& R5 ^ClassNumber(Q5) ;
ClassNumber(M) ;

PicardGroup(M) ;
( q! l3 ^$ R! b3 h  E: DPicardNumber(M) ;

4 @' ^5 k, M- \0 u6 y; F+ B
3 r3 ]0 M/ ?- ^) q% L, C8 y$ tQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);

" u# [7 ?" ?6 g, r
NormEquation(Q5, 5) ;
" K: O. X% t3 iNormEquation(M, 5) ;

/ I- f- a7 |% N  c! \4 b4 x
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
/ u+ x/ \; R9 ~$ N1 j  FUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
& t( ~+ J2 |9 K: A! @$ x6 gMaximal Order of Q5
; T1 ~( u* d0 P: W2 \Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
true Order of conductor 16 in Q5
7 U' j5 l& F2 Htrue Order of conductor 625 in Q5
1 }" w: l9 X& T; \/ ~true Order of conductor 391736900121876544 in Q5
[
+ u5 d3 X. r) U3 S    <w^2 - 3, 1>
. R' b* |; U5 b/ q, b* X( x4 F4 H]
5
1/2*(-Q5.1 + 1)
-$.2 + 1
$ X; [5 V) U- V  ~5
Q5.1
$.2
4 t/ d8 w# g$ n) I  S1
3 T1 r0 [0 {  O$ s, aAbelian 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
) E* P" o0 U. e* Z  U2 x' u1
1
Abelian Group of order 1
' k- H9 c# k/ i7 RMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
; {4 W* |7 E+ B' w3 b$ u/ U; k/ minverse]
' a+ \6 M1 ~$ k; s1
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]
Abelian Group of order 1
1 ]: u8 B5 V" y7 l- g& a5 mMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
/ t, q/ ^) P* T. B1 q0 ]) N5 given by a rule [no inverse]
% N# _; n5 U' f0 t" g+ `6 V* r' ctrue [ 1/2*(Q5.1 + 5) ]
true [ -2*$.2 + 1 ]


, M+ v. N; D3 K4 x8 d2 Z* l+ h5 N

" x5 W" I8 f+ N: h, Q7 W
. @( \+ b1 z- ^: Q

, H8 E2 V$ P4 |" L0 t! y8 C  m
/ ^6 A# V' Y" h% w* ]' K; J+ h* U

6 l* B5 S3 o7 V/ }
==============

Q5:=QuadraticField(50) ;
; Y8 j/ U1 @# `Q5;
4 L( }- _2 D- [! ]( x5 [
% r6 c, V4 H' a3 J1 a" lQ<w> :=PolynomialRing(Q5);Q;
$ c6 F9 W0 O7 R, M9 |. X0 ^# _EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
' P8 D, s: g: v0 J8 MNumberField(M);
. P' C2 J" h& u2 _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;
IsQuadratic(Q5);
) _# x% w$ a  u& OIsQuadratic(S1);
1 h; L2 g' G+ {2 yIsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
; ]0 G! `0 p, I# V9 uFactorization(w^2-50);  
" a3 ?7 X: d: {" ?Discriminant(Q5) ;
# p; [4 m' I' x) B0 OFundamentalUnit(Q5) ;
6 L) |) D" q1 v/ UFundamentalUnit(M);
& x& b7 d% B8 m. O8 IConductor(Q5) ;
1 C1 y1 j) i) H' l: r
# {, V! K6 k$ D  z% _) Q1 l/ kName(M, 50);
) g& q' W4 p0 d/ bConductor(M);
ClassGroup(Q5) ;
9 P0 ~$ J8 g0 ^4 ^8 E% |% }8 nClassGroup(M);
5 t9 K8 f8 N6 rClassNumber(Q5) ;
( Z& B. Q( I0 u4 RClassNumber(M) ;
2 _1 [' F7 W" @3 ~- BPicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, 50) ;
NormEquation(M, 50) ;
" n& y& {$ ~+ [3 w7 l
. ~# h: o) ~; m9 x- f$ z& w0 tQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
5 y" C& D. C0 f! g0 ]; bEquation Order of conductor 1 in Q5
% {9 l5 p2 T. S3 zMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
- V5 @: ~6 ~( D; X0 I1 x! _Order of conductor 625888888 in Q5
' g( f" w9 E3 q5 ~true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
6 `5 ^" g/ F" A, K+ ?% P# H/ V6 `2 Ttrue Maximal Equation Order of Q5
3 \. y5 Q  z; \: l0 M* ytrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
( w% {% F! T3 \4 g, k+ h[
* {8 g5 s8 e7 m) Z. W6 q    <w - 5*Q5.1, 1>,
2 E! p4 W  ]( w5 D$ K    <w + 5*Q5.1, 1>
- T- v+ H, y6 C. F, G# t  r]
8
: B; O8 {, |; M0 ], mQ5.1 + 1
0 |8 P( q- S4 e. D+ M$.2 + 1
% N" v; V4 ^! C$ z5 ]' X' N& k8

) e/ d7 C5 S* \  k- S7 i9 [>> Name(M, 50);
9 [% {& e0 ]( _3 U, l( s6 d( ^+ h6 I       ^
' P9 D- I+ d( c4 _4 m; m: k) ]Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]

1
Abelian Group of order 1
  Y% L1 h/ F, Z/ u9 W: S, _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
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
- i8 Q  q' t: g: x9 ]' a1
2 W) e6 v+ r' w' G" }Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
$ k; ^5 z. J0 M7 \7 G8 h7 G9 A. F8 given by a rule [no inverse]
Abelian Group of order 1
. m; E; u" D0 z# B3 v9 _+ c3 `Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
true [ 5*Q5.1 + 10 ]
& ?3 Z3 V4 B3 o* ?true [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

8 v: T( q9 L2 A, m- U0 ]基本单位计算fundamentalunit ：
5 mod4 =1                                              50 mod 4=2

x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.
  B4 C# N0 k& p, ~: F! o* y0 M3 `; @2 ?
% t" z' W7 f. _9 h
4 e1 n5 u: N5 @  Z0 t1 t. V最小整解（±2,±1）                              最小整解（±7,±1）
                                                             ±7 MOD2=1

两个基本单位：

lilianjie1

lilianjie 发表于 2012-1-4 18:31
2 g, E: J& H. `0 S8 I基本单位fundamentalunit ：
5 mod4 =1                              50 mod 4=2

5 ^" U, M, p( g/ W基本单位fundamentalunit

lilianjie1

. C* p! M8 [8 G- S判别式计算Discriminant
! w  H0 L4 u% M. T; l
5MOD 4=1
0 k# ^8 a5 ^$ \- K1 ?& q
（1+1）/2=1          （1-1）/2=0

, J2 \$ M" u) \: b- nD=5
' ^  G8 a( z+ ]5 f0 P) N! R/ U4 b
. A& D$ T6 l; b3 S: ?3 g
" b+ v5 I4 K8 T4 \7 Z50MOD 4=2
+ ^3 k( @$ Z2 qD=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;
% c1 n0 _0 I) V9 I. gCyclotomicPolynomial(5);

& x# _, d5 x" f) D) S$ k$ T
分圆域：
! H7 g$ S2 c! P. M; }" a分圆域：123

R.<x> = Q[]
F8 = factor(x^8 - 1)
F8
, }: Z: [, G# Z: J" y9 h/ B% a
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

1 y" L2 i# P" ]/ o* d7 l3 T8 E( aQ<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
, i# l) O; }2 V: I, yFF:=CyclotomicPolynomial(8);FF;

F := QuadraticField(8);
$ J9 p2 T5 x% t& |8 f1 PF;
D:=Factorization(FF) ;D;
3 {/ j4 q. w% _* FQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
2 J' J4 J4 }) _/ s+ jCyclotomic Field of order 8 and degree 4
$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
6 S1 I' I2 E5 h) E0 n8 V- I    <$.1^4 + 1, 1>
# w# c. J8 M* ?  @6 g6 o. m]

* S6 k* k% R! zR.<x> = QQ[]
5 u6 P% Q# L# f8 R: g" AF6 = factor(x^6 - 1)
F6

# f. S/ z6 Z* Q& h, f, I6 H(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
' p4 S: b8 d% k# Q0 ^5 Z
Q<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;
6 B! e- E5 ?# Y6 [  L/ z. T
, `3 }( @1 g3 G& d& Z  n6 qF := QuadraticField(6);
F;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
( J, l$ w6 B7 K8 u, o4 \" V' n$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
    <$.1^2 - $.1 + 1, 1>
]
' |4 u. }; ^; l! P2 j5 k+ p
R.<x> = QQ[]
F5 = factor(x^10 - 1)
F5
4 Y8 V. B2 {2 S+ e(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)

% r5 ?* }1 D: R; `6 OQ<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;
4 C% m& A5 |; O1 I% F! w- U- P
F := QuadraticField(10);
7 {4 ]3 e% X$ ~6 yF;
4 d; U% w9 f. I% K6 m! y: \: eD:=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 - $.1 + 1
9 y  c' J( W) X% d$ c9 v& Y* q; WQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
6 v$ o. i4 I0 P4 @* c! x[
9 F% N, x5 I: J1 _. i3 ?  P$ @5 W    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
" l5 n& p  U! G. j: Q6 a]