实二次域(5/50)例2

lilianjie

1 y  m! e) _  ]: K" R
7 `+ U0 u. Y2 ^+ |8 d0 P2 c7 xQ5:=QuadraticField(5) ;
Q5;
, Z) `& A5 q$ ?Q<w> :=PolynomialRing(Q5);Q;
( |% a7 L( v+ G7 ?: T+ y% |5 F
8 N! J+ c- d0 l& QEquationOrder(Q5);
1 @; {8 Z- w8 [3 VM:=MaximalOrder(Q5) ;
M;
2 ^1 D  o/ g+ @. m0 V* J6 xNumberField(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;
, L5 o. b% ^2 [) {0 q% qIsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);
! r* w& R& E: L# F2 [  h: i9 X2 u& }Factorization(w^2-3);
Discriminant(Q5) ;
$ J& ~: L' r) B2 m/ ]. j7 F, pFundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
Name(Q5, 1);
Name(M, 1);
8 G: V2 ]& Q& d# SConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;

8 G9 b6 x2 r( \9 @+ g. ePicardGroup(M) ;
% u" J; j" |# ~& P1 FPicardNumber(M) ;
  J  P( ]6 F9 y) s/ c

QuadraticClassGroupTwoPart(Q5);
- g0 {* H, S2 zQuadraticClassGroupTwoPart(M);

: x. r5 w4 e! H) r5 @3 ^" P% J: C
" A& [+ W1 X; Z* W  z$ wNormEquation(Q5, 5) ;
9 Y$ D, F/ h5 _' q2 BNormEquation(M, 5) ;


Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
1 T' S+ N5 W) |4 B2 N$ ~$ b; [Equation Order of conductor 2 in Q5
Maximal Order of Q5
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
/ O& T/ u+ R* Y2 JOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
% i# p9 v$ f) c9 w$ Ktrue Order of conductor 16 in Q5
( t' }, f5 |' P4 S/ @9 ltrue Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
[
    <w^2 - 3, 1>
]
5
9 o% @: ?, Y2 Z$ k! R5 R; e( N/ w# J1/2*(-Q5.1 + 1)
-$.2 + 1
5
Q5.1
$.2
3 D9 w- o! v. D6 I- P1
! k/ N& q# x+ b0 mAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
1 {( G9 ~4 L: k" x" n4 g  j9 R1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
* k4 K3 W% H, g+ N4 `2 ^1
Abelian Group of order 1
( t. q: J+ {0 v; E4 Y4 BMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
7 U( z: r5 C6 W5 R/ J$ J+ T) O; yAbelian Group of order 1
* Y& T; M& _8 L. O) iMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 `6 [8 V+ I( @; }$ A' O5 given by a rule [no inverse]
: Q4 L5 m0 W! o; G: x) Ptrue [ 1/2*(Q5.1 + 5) ]
true [ -2*$.2 + 1 ]
- L1 H8 _; O  F7 Q9 S' r" L/ N
% U# V7 p/ k& G6 f5 e
6 v3 X5 N+ Z7 d8 P4 Y1 |
1 W: c  p- }' c) A/ O

4 I; L% w+ C7 k, g
9 w; C  J6 n5 E. Y) U' N) e- E* s; _

8 Q$ S' C( W& B! A% d2 I

. \1 a9 z' O# e' \# b% q( f) N5 o
2 ~4 i( S# j% h7 h' o8 Y==============
- L5 c9 @  h. C8 O0 ?7 I
# d" S5 ?6 Y! g9 T$ |Q5:=QuadraticField(50) ;
Q5;
2 I  q# {; y# n" p5 z6 u) U+ J
; T' V+ t- j2 D" J! _" [Q<w> :=PolynomialRing(Q5);Q;
6 D. U3 w9 @2 w- Z% ]( ZEquationOrder(Q5);
  M# T2 |, g7 F, @M:=MaximalOrder(Q5) ;
; J0 T  H9 h% ?* MM;
NumberField(M);
: M! Q7 k2 C1 K; d& j& sS1:=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 c7 p. ^% E$ q2 j: o: \7 q& \, cIsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
& x4 a5 n8 y4 ], h4 G$ WFactorization(w^2-50);  
) G7 ?' Y( H8 Q5 t: N( fDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
7 |; T3 A( U1 Y' C( sFundamentalUnit(M);
; p1 X: l; _+ N1 c' c, lConductor(Q5) ;

Name(M, 50);
- @5 h1 L: ]* a$ K# h. E* BConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
/ B* ?2 u5 j, n, n. d: VClassNumber(M) ;
PicardGroup(M) ;
8 R4 {! o! r5 zPicardNumber(M) ;

+ I  w. }6 F4 R4 X' `; kQuadraticClassGroupTwoPart(Q5);
& O5 P: y6 c3 B1 H2 H. [QuadraticClassGroupTwoPart(M);
. s" g1 N  I4 b" I' RNormEquation(Q5, 50) ;
NormEquation(M, 50) ;

) e+ U1 c4 i, ]Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
3 y9 x  F( y8 wMaximal Equation Order of Q5
: E5 ^6 K! f! Q  c% o# m( K$ |9 C% CQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Order of conductor 625888888 in Q5
6 D2 _# G1 Y7 _, {- @true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
true Maximal Equation Order of Q5
/ B' [) x- A* Jtrue Order of conductor 1 in Q5
+ c, p5 m+ F5 k/ ltrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
0 q/ Q: @0 T& h1 ]4 m* Y    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
( j4 B& V* O4 J( J6 L8
; b6 H. b2 C2 }Q5.1 + 1
% ^& I  w# g' m; M6 q. K$ ^2 Z$.2 + 1
" M( ?* D& |  M# _8
- T2 L  f* C# }) {& H' k
>> Name(M, 50);
. G6 k& f6 {. A# S+ v, i5 C- F" S       ^
, K, `8 A4 B7 o# a! h5 r, nRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]

5 d7 c9 J$ u" i- Z$ V9 l7 `: e1
" `/ U- ^0 Q; r  i0 C( L3 D4 u* dAbelian Group of order 1
( z' d8 j  e  t: L8 p6 ?$ a% W2 @Mapping from: Abelian Group of order 1 to Set of ideals of M
/ H3 V! J$ l5 m5 c5 U+ O. o1 jAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
% p- C3 x3 D, J7 G0 H: _8 H1 \: T1
, x& S6 [5 Y* k5 ^* n1
% F& h: ]$ ?- P: H% r* uAbelian Group of order 1
- I3 A. v- [; s$ t0 wMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
( `1 ~4 t, \: H* \inverse]
  C+ o. j) T+ k9 x& d& ~# D* _8 `1
" t! y  V' ^# _) G  w) TAbelian Group of order 1
5 R5 G5 J) `5 ^% ~% ZMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
- ~. Z. @: W) P( ?" Z2 n/ Q& ?Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
. U. S- B( ]  strue [ 5*Q5.1 + 10 ]
$ r- X1 P7 l7 N  Xtrue [ -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

: u0 q& J7 F- o- P$ T& D: c; \) w x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
  ^0 W4 G: N, D* E8 p' m# B- d" W x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.


最小整解（±2,±1）                              最小整解（±7,±1）
6 H0 l; [8 |8 `9 p0 n8 l5 I! O+ ^6 I                                                             ±7 MOD2=1
! V& |7 y  H4 ~& B4 i' I5 ?9 q) R
! s: o! Q" w6 O" ^' n( k; s两个基本单位：

lilianjie1

lilianjie 发表于 2012-1-4 18:31
1 T# P( O5 m/ b基本单位fundamentalunit ：
5 mod4 =1                              50 mod 4=2

基本单位fundamentalunit

lilianjie1

判别式计算Discriminant

3 Z5 f" d+ B% N+ ]5MOD 4=1
! H! n+ C: o& D, w2 R
5 L$ n& S' W! b! N3 K（1+1）/2=1          （1-1）/2=0

! O; X$ u/ O+ n  r* L$ h  sD=5


50MOD 4=2
! V# x, V1 j' u7 U7 MD=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44

: e) [3 l7 w, [$ g+ B( j. n4 G* `分圆多项式总是原多项式因子：
6 e0 r' w+ @9 AC:=CyclotomicField(5);C;
- [* j9 M6 J( m; `& `/ H( OCyclotomicPolynomial(5);

4 B- Q7 g0 d5 W; F& Y分圆域：
1 C$ @) M1 T7 K; T) `4 d分圆域：123
% O0 U$ E! Z$ @8 _! P% v; @/ j$ x4 k
R.<x> = Q[]
; w) D, f1 h1 p# v/ JF8 = factor(x^8 - 1)
F8
: ~2 ~- i* `$ }* p* w
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

# I# O: x1 T' pQ<x> := QuadraticField(8);Q;
' l/ h$ @- E8 g9 A( S+ KC:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;
8 g) V. s: Z5 e0 s. }, m
F := QuadraticField(8);
/ t; k' e0 O4 AF;
' H4 {  B2 f2 l5 R; r6 J9 VD:=Factorization(FF) ;D;
$ @3 C. D1 S+ n! L9 {& r& cQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
9 b* @; q7 b* d/ {! XCyclotomic Field of order 8 and degree 4
8 h8 P+ x; K$ |3 n9 a9 s$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
; t1 X( V5 ?+ G( C9 w[
, S2 F5 K2 @4 A1 i; U; C    <$.1^4 + 1, 1>
2 ~( J) R0 w; j1 C( N& d]

8 y/ @" Q- ~& F3 KR.<x> = QQ[]
F6 = factor(x^6 - 1)
3 u& F- o2 B# ]" Y# ^F6

, H9 M5 k7 |! r(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<x> := QuadraticField(6);Q;
' K' Z& Q1 F0 WC:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;
1 v+ ], _3 P( D8 O- K: y
2 x$ }% N! K; Z/ Q/ IF := QuadraticField(6);
+ l4 w/ P5 R6 A$ K3 s8 EF;
D:=Factorization(FF) ;D;
  F. E" z( R8 Y1 ?% T! FQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
% P9 W  f3 t7 o4 g- W6 iCyclotomic Field of order 6 and degree 2
! t* P7 [# W+ T7 h# N$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
8 x0 J  q' j" f9 Z( ^) J1 a# ]2 H[
& y- L& u' j# Z, k5 M    <$.1^2 - $.1 + 1, 1>
8 d* d" p4 O, f* D% q1 x  o" h* y]

, G4 A, X8 v4 w9 t/ i- SR.<x> = QQ[]
F5 = factor(x^10 - 1)
& T1 h& ]* \" V  y5 r) `* C, jF5
(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)

Q<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

; c, o; I( B9 w7 n; cF := QuadraticField(10);
F;
D:=Factorization(FF) ;D;
% R& Z/ A0 \6 IQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
4 \$ j9 Z! A( d& y( f$.1^4 - $.1^3 + $.1^2 - $.1 + 1
' n9 r4 [8 L4 a0 P' }. x1 N# w  c' VQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
3 ]; h+ q: {" f7 u+ R+ }[
" I7 U) W; j7 F0 Q    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
# D. X. N; m2 x: k2 D# x8 _1 Y]