实二次域(5/50)例2

lilianjie

, ^  S* x% W& T1 W, d  I6 U
Q5:=QuadraticField(5) ;
Q5;
Q<w> :=PolynomialRing(Q5);Q;
! F* C$ F6 M" L
/ o- E) B/ K) n3 EEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
  p9 v. N2 Y+ ~9 j/ u$ J( zNumberField(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;
7 P* {0 y% Z- S( \' P7 kIsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);
Factorization(w^2-3);
Discriminant(Q5) ;
# e* ^. ]1 G$ o7 \# Y% gFundamentalUnit(Q5) ;
FundamentalUnit(M);
+ Y, m9 A- k+ uConductor(Q5) ;
Name(Q5, 1);
/ q# H! R( k2 k* `) Z* L- l+ rName(M, 1);
  {0 ~/ e+ |' V9 v5 bConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
, D0 s3 _, t) e3 N& d! J2 H& wClassNumber(M) ;
- [. d/ k- A/ ?+ m+ o3 p
PicardGroup(M) ;
PicardNumber(M) ;

; [# X4 F" t% ^- T3 N. ^
/ x. J9 U  s3 I' P# z5 H# PQuadraticClassGroupTwoPart(Q5);
7 g( h3 a. i$ D- u  a( E" U1 UQuadraticClassGroupTwoPart(M);
0 A+ B9 x$ n% B' E( X+ H) {! S
5 p/ M$ ^$ c- |- A' @& p
NormEquation(Q5, 5) ;
NormEquation(M, 5) ;


Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
& p2 e3 ~# R6 x  w7 bEquation Order of conductor 2 in Q5
7 j9 a$ L. q7 K3 @' E4 S% zMaximal Order of Q5
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Order of conductor 625888888 in Q5
' @$ Q  V; o7 T& }9 q+ r9 K3 {2 Itrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
true Order of conductor 16 in Q5
true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
[
    <w^2 - 3, 1>
2 j- G: r! p# |; o( d; R" D]
5
1/2*(-Q5.1 + 1)
-$.2 + 1
4 k1 Z* ]1 S. R. j5
" o  D+ Z7 G% n" M% l1 l, ^0 ~Q5.1
4 o5 h5 d. k9 X& r# `! j9 r$ t$.2
" j! q0 o, _, f/ l1
. W1 ?" Y# L$ R; [( ]Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
# T& a. _; Q: \, H' pAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
7 p* X9 L' s) Z" q1
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
! A- M3 I0 d7 ^% dinverse]
3 \  S( g& s: N/ p& s: D% }1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
- G* h" [" D9 z  i5 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
9 L8 ~7 w1 g) d5 given by a rule [no inverse]
true [ 1/2*(Q5.1 + 5) ]
( k8 B8 ^  X3 T% Jtrue [ -2*$.2 + 1 ]
* n6 {3 w6 F& v1 o/ Z: o
! g( j, |. S3 b) }. u7 Y! p

" b( w& Q/ [2 r3 S




, {# J) \1 s2 U" k
( M3 V; B9 [' v/ [  u
! Q. y6 D2 x5 i5 v: W( P
==============
" C+ M, i" Q7 m; ~3 S2 h, R( F
% B! D9 x9 Y1 DQ5:=QuadraticField(50) ;
, `  b9 F: C; ]+ JQ5;

8 A- l) f! l* u  y2 CQ<w> :=PolynomialRing(Q5);Q;
5 O7 U* Z, p! s; ~. h0 LEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
+ R' T$ j8 Q' b/ ~) z  [2 o$ gM;
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;
IsQuadratic(Q5);
' i/ W. }# y; ], x; aIsQuadratic(S1);
IsQuadratic(S4);
- _2 ]8 K! \/ Z# ^IsQuadratic(S25);
: b  P5 @7 H7 r9 FIsQuadratic(S625888888);
Factorization(w^2-50);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
, ^* K( o$ H/ J4 ~* L0 v  KFundamentalUnit(M);
0 `, O. Y' \/ b6 l; mConductor(Q5) ;

$ `* p( [0 s. PName(M, 50);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
% q& P; X. t- y) rClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
9 o% L" \+ G$ W2 C! V
6 ^0 Y' b% w9 z$ ?% N* O. oQuadraticClassGroupTwoPart(Q5);
8 G3 |* z! ?6 rQuadraticClassGroupTwoPart(M);
( d- W; _; b$ _2 }NormEquation(Q5, 50) ;
NormEquation(M, 50) ;
/ ~3 M. y$ E5 M
+ s+ ~  y$ k% Y% F) \Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
/ R0 n/ o) V% A% a; |7 b$ H$ cEquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
" N/ u7 g6 O1 R7 t. d/ |$ @Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
* c7 S1 N% X5 w  o+ k. t  }" R' j: V[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
: U& n5 B  Z1 Y4 a5 j1 e. Z8
Q5.1 + 1
- K$ v2 Z4 @' q+ ]0 E6 B$.2 + 1
! \0 ]* J, P$ V$ p+ y5 e6 s! F4 a/ j$ i8

>> Name(M, 50);
8 a; p% A7 Q" h) c# ^) ]" j, h3 T       ^
- V/ `) ]0 p3 Y  ?" F' ?# IRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]

1
Abelian Group of order 1
# V6 M, Z& N; L/ dMapping from: Abelian Group of order 1 to Set of ideals of M
: i/ ~$ J, m5 d0 ZAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
% l7 C& n: e7 E9 l5 T; R8 e6 {1
1 V( ~1 _' g$ \' `7 Y$ s/ R1
' r; _3 a- H, s# a+ X& U6 jAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
4 l/ M+ H" c; o" dinverse]
% R6 x& V0 V0 n# V0 Z, `. }6 `5 \1
9 P6 ~7 x9 ?+ x; qAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
7 w1 }' R( t$ p3 v8 given by a rule [no inverse]
. J' F0 g% v0 T' Y  bAbelian Group of order 1
9 m% I/ J' ~% ?/ B4 X4 TMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
! H0 c- {1 D0 r, o1 g9 i  g8 [% I& u; |true [ 5*Q5.1 + 10 ]
! K+ G6 u% }2 Y4 {6 xtrue [ -5*$.2 ]

lilianjie

lilianjie 发表于 2012-1-9 20:44
! N' B) q7 [- l+ F0 _
分圆多项式总是原多项式因子：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

9 Q/ X- ?" D2 j; }$ d
- G+ n  h( @% y( O分圆域：
分圆域：123

/ X# q) `" i' b. MR.<x> = Q[]
+ E- K/ X* q2 v( S/ _0 E. PF8 = factor(x^8 - 1)
) `+ b2 e% ~5 l3 ^4 |F8

(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

Q<x> := QuadraticField(8);Q;
+ F: Q6 Q& d% _" ~" xC:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;
3 p0 _  ~* R, ]' @
F := QuadraticField(8);
F;
D:=Factorization(FF) ;D;
: o6 T, Q8 c- fQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
* z  E" ~! G, T1 v$.1^4 + 1
- S! D1 ~  A# E9 m: XQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
7 h3 L, c+ E, z- c+ t8 }; `9 K: Q[
* w9 {8 b8 ?3 z% g7 I& P# Z! }    <$.1^4 + 1, 1>
]
7 {) P% O4 I9 ^
R.<x> = QQ[]
* ^6 M% v2 q$ t$ i4 S) t* cF6 = factor(x^6 - 1)
: ?# ?/ A/ r/ l3 H' s4 VF6
8 X1 K2 H& a# H& b* D/ j  n4 a: J: f! H
6 k6 c: Y& [7 L7 j, |(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
' q, h/ \" s3 e8 y% z
Q<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
) b; v7 f5 Z5 P! GFF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
. |$ v# d% |+ UF;
( J! p! b9 V# a3 k$ cD:=Factorization(FF) ;D;
3 p# V' R: h8 W0 T7 T4 }Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
) Z1 O# d  @8 W$.1^2 - $.1 + 1
' A# V% \6 t2 F9 ^Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
    <$.1^2 - $.1 + 1, 1>
' w2 R$ r6 O% m. ^]

R.<x> = QQ[]
F5 = factor(x^10 - 1)
) s4 ]3 Q' S+ T' W  `9 Q" H5 |/ d3 i8 ?F5
/ ~' ^8 S; D- r0 O' v(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
" R, N5 U1 q: b& R2 I9 X$ m1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
7 @/ n7 u% p6 _  ~9 M' r
Q<x> := QuadraticField(10);Q;
+ I- t; x; t9 ZC:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

F := QuadraticField(10);
F;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
4 u" `* u% b4 PCyclotomic Field of order 10 and degree 4
3 ~6 X" L1 M% R3 I4 `9 x( e7 ~$.1^4 - $.1^3 + $.1^2 - $.1 + 1
) M) K" X0 a6 G* J( uQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
2 x7 t8 K7 h( [. W( S& y    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
+ l  `8 w# b; N' {7 Z4 o) X]

孤寂冷逍遥

lilianjie1

3 o2 _" p, ?. }  a判别式计算Discriminant
: A4 {* u2 \& Q2 c/ G1 V/ |  t% ^
5MOD 4=1
0 v/ B5 h% c- D% i9 V% r6 Z9 i, E
! E5 s; z+ {* |  m: _（1+1）/2=1          （1-1）/2=0

D=5
6 _' Y8 H' G# b9 U& s
( e& }( w7 p  N
50MOD 4=2
D=2*4=8

lilianjie1

lilianjie 发表于 2012-1-4 18:31
, e& h1 }% m5 w( a5 j: M8 ?( w' G1 Y7 C' \基本单位fundamentalunit ：
  ^& |, }, u7 W5 mod4 =1                              50 mod 4=2

/ u# C/ E$ q: U: T( I. W* ?( `% o9 ^基本单位fundamentalunit

lilianjie

基本单位计算fundamentalunit ：
' b5 X; A: ?- \; s7 y) ~5 mod4 =1                                              50 mod 4=2
3 m+ D( `' O8 l! V' \" Y, @
x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.


最小整解（±2,±1）                              最小整解（±7,±1）
  M! Z9 S( @! R                                                             ±7 MOD2=1

两个基本单位：

lilianjie

二次域上的分歧理论