实二次域(5/50)例2

lilianjie

- S+ {& y, X3 \# L* h# qQ5:=QuadraticField(5) ;
5 l8 L& M7 f% O9 JQ5;
Q<w> :=PolynomialRing(Q5);Q;

& b" ~) \  I, I5 ZEquationOrder(Q5);
# I0 A5 y! x4 M1 _( OM:=MaximalOrder(Q5) ;
M;
NumberField(M);
6 W! |2 `9 N1 n  y; ^1 MS1:=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);
/ H' ?( g* }0 ?. }Factorization(w^2-3);
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
$ V1 ~1 [- c  HFundamentalUnit(M);
Conductor(Q5) ;
Name(Q5, 1);
/ i$ z, F. n4 k1 bName(M, 1);
Conductor(M);
4 |  D, A5 Z6 a$ R; M) e; KClassGroup(Q5) ;
ClassGroup(M);
5 A! A3 s- T% l  }+ gClassNumber(Q5) ;
ClassNumber(M) ;

  O- k: y. Z# P1 [, CPicardGroup(M) ;
PicardNumber(M) ;


1 o) [: U" ~, ~# \6 S% B& M9 jQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
: B; {4 s& v' x6 `0 Y4 e) k7 j

1 n, l" p% U& J3 F% H. k7 mNormEquation(Q5, 5) ;
: J2 f' o  x" A& v3 b0 \5 x- K/ f8 BNormEquation(M, 5) ;


Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
3 z; Q* e; f7 MUnivariate Polynomial Ring in w over Q5
) m( K, ~4 L# O# G  C3 @Equation 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
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
true Order of conductor 16 in Q5
true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
[
    <w^2 - 3, 1>
]
5
  ?6 T+ j7 L# \8 m, s, W1/2*(-Q5.1 + 1)
7 n1 ]* v' v* ^6 _* [4 u+ n( c1 g8 O-$.2 + 1
5 j$ z8 H/ N1 e5
Q5.1
  s; X5 P8 C# z9 r) k) ^$.2
0 o2 J, K7 e3 c2 D+ I1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
8 |) M/ F! ]& E3 Y, N1
1
Abelian Group of order 1
- X% Y' }' A4 b" L" m6 EMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
2 n; z( D% e$ R7 W* c  y* [: \4 ?inverse]
1
2 w. j2 Y  o: _4 b' k! MAbelian Group of order 1
! i6 j) s; B% W+ _. }) BMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
3 p2 I; s, e9 _% e* mAbelian Group of order 1
) r; i8 b  z8 C0 N/ ZMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
% [$ H, u9 s) c) ]1 h. `3 ytrue [ 1/2*(Q5.1 + 5) ]
$ v1 B5 ]# U+ @3 N$ p6 O7 V; @true [ -2*$.2 + 1 ]
/ n) P+ u0 Y3 p5 T7 J: U" S



" K& W4 b$ w- [7 k
- M. ^- y* n( a. e+ p
% K! m3 ]. s3 M& k' `# ]% X
- g4 a) m5 r$ k1 d

+ b; p; v* q. B( F( t+ P1 N

==============
& n! R1 @, |9 E5 ^, h
Q5:=QuadraticField(50) ;
Q5;
- [! F+ |; M) I. j) V8 \# b" n1 p
Q<w> :=PolynomialRing(Q5);Q;
" ?. ~3 F- E- N/ Z  hEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
+ ?& z& v& f" z' {7 RNumberField(M);
0 J2 p* M+ n8 I! x  F9 [8 a' _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);
IsQuadratic(S1);
" z" ], `: l$ M& w& K2 K1 YIsQuadratic(S4);
IsQuadratic(S25);
8 _) s/ I' C" z8 xIsQuadratic(S625888888);
1 ~/ l8 r0 `1 X- I8 W2 o) Y& D7 yFactorization(w^2-50);  
Discriminant(Q5) ;
. t% }& X4 A6 \' n2 yFundamentalUnit(Q5) ;
0 g. c9 u7 |4 L( ?! \) {- oFundamentalUnit(M);
Conductor(Q5) ;
' r7 d0 K) Q" W
. ]; ?8 r1 F1 f4 ^Name(M, 50);
Conductor(M);
: h0 h$ y. ?6 aClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
5 S% N3 o6 p/ s  gClassNumber(M) ;
$ q+ \1 y$ }$ b5 t' Q+ ]- oPicardGroup(M) ;
PicardNumber(M) ;

! X7 e4 D  d3 i$ w% ^2 GQuadraticClassGroupTwoPart(Q5);
  x4 K3 t. |; i$ G$ {/ HQuadraticClassGroupTwoPart(M);
  x. R" ]$ H, \: yNormEquation(Q5, 50) ;
NormEquation(M, 50) ;
6 p$ C! `/ p. N0 F2 Q# W' s
5 ]; o4 r5 F2 E0 T( sQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
3 f7 H1 P* s. m; RUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Order of conductor 625888888 in Q5
; @- M& i* @# \true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
" I2 j9 M5 X+ @! U% n! \true Maximal Equation Order of Q5
1 }! f; |9 ^7 ctrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
; l! o3 H* R+ A5 }2 D[
    <w - 5*Q5.1, 1>,
: X' P) ]  h* x" ~2 h    <w + 5*Q5.1, 1>
$ i# N  q; P& i% U% u]
8
! _% W$ S4 r6 Z* L; \4 JQ5.1 + 1
$.2 + 1
8

>> Name(M, 50);
) `. S* ^9 X  z; L8 c  P       ^
Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
9 }# M) L5 I: J; G
- i/ ?. f, O1 W1
- l6 T' o$ P  l' f0 G7 ~% vAbelian Group of order 1
5 Y( u7 R. e8 L' m0 YMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
. O7 |3 e7 k9 a, vMapping from: Abelian Group of order 1 to Set of ideals of M
7 d( X* q5 i9 `; Z2 |1
1
Abelian Group of order 1
$ [/ \0 m5 g( g$ N2 ]: g' a1 J2 X' tMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
4 p# w4 w) B0 F8 U; C/ H$ Zinverse]
' l; o  `4 V( ^/ u+ P1 P) T1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
% t4 w, e6 j& `. d' n; e2 v8 given by a rule [no inverse]
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]
6 N4 I" B* n4 I8 @+ H- Ftrue [ 5*Q5.1 + 10 ]
true [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

% P2 S' `2 v0 R9 r
  [; ^  d# y2 I) K7 o3 w( v基本单位计算fundamentalunit ：
3 T, x, g, `) W- }/ [5 mod4 =1                                              50 mod 4=2

, e! j0 w8 s2 u0 T x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.

8 A$ \( @' `/ Z
最小整解（±2,±1）                              最小整解（±7,±1）
                                                             ±7 MOD2=1
3 T- p9 S& M; B$ G, p, j) ?
3 B2 B+ m, {# b/ c2 O* W6 m0 F. i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

2 H7 k$ c0 H" P0 a  G判别式计算Discriminant
6 I* p8 g+ j+ [& |
: q2 h# |5 e) Q& d5MOD 4=1

# m& C2 M) w! t, K（1+1）/2=1          （1-1）/2=0

* b8 d" y6 |, a( P6 D& JD=5


) i" f! C. U% T  z7 I50MOD 4=2
( d! T$ B9 Y0 j* zD=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44
% n- X. u8 k  f2 K3 w- y
分圆多项式总是原多项式因子：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

分圆域：
分圆域：123
) L, _  t4 a, M# P& \
R.<x> = Q[]
' K( r# G# ~  {/ O  R; `F8 = factor(x^8 - 1)
3 l( t: a: x3 W1 n" e7 OF8
4 |- q/ K) r( f4 ]2 a7 m
# h% R4 m% m" e2 }% ~# F(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;
C:=CyclotomicField(8);C;
: p5 d; `! L) f) p: w6 e( `' oFF:=CyclotomicPolynomial(8);FF;
& U: J! o( o" l9 z( r/ \1 k. e$ A
/ ?; U% r$ k3 ?# L+ lF := QuadraticField(8);
  R, C* [& e! [, E3 J3 bF;
/ k' p6 M3 @: g6 z0 P: U5 K. z, LD:=Factorization(FF) ;D;
/ y: v+ r6 ?4 n! \8 O. UQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
0 f; f, x% |3 k( P* M3 VCyclotomic Field of order 8 and degree 4
$ Q. u6 B$ n. D9 P. a6 k$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
    <$.1^4 + 1, 1>
% C) Z) }( r1 E4 o( Z0 M]
7 ^% w' w; `+ W" z. d8 H
R.<x> = QQ[]
F6 = factor(x^6 - 1)
3 P* T7 ]& s3 {. f) QF6
& ]' k4 f& \; d
(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;
C:=CyclotomicField(6);C;
6 D: D$ N: X8 j' Q  \FF:=CyclotomicPolynomial(6);FF;

; J" @+ x9 c+ N7 kF := QuadraticField(6);
F;
* Y/ B: n) J' aD:=Factorization(FF) ;D;
5 w" D: q8 W1 rQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
# ?' X7 \" f7 i; |2 LCyclotomic Field of order 6 and degree 2
5 z( a" C, Y$ E- O7 [$ h$.1^2 - $.1 + 1
! ~( k9 S* b; @& D) O$ Z8 o0 nQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
+ M( R1 C2 U9 }  l; _2 O5 d[
    <$.1^2 - $.1 + 1, 1>
. k" s. {" q9 C]

8 l# X6 j* z0 N8 L, hR.<x> = QQ[]
) |: H* L) [- W  x- h  mF5 = factor(x^10 - 1)
F5
& a2 }; U9 d  s8 k1 q(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)
' t2 R& K  B5 h* u6 _1 I0 g
. k! K: s/ b( O& t& V2 eQ<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

, y; ]5 m. z0 uF := QuadraticField(10);
0 Z8 {7 E% a+ M5 d: |" rF;
# ~( E. j' Y( P& N$ WD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
- a" b  {: S2 R, r3 l$.1^4 - $.1^3 + $.1^2 - $.1 + 1
" C( V: \7 N; E3 p  Y' vQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
4 g6 J: l  n# {* ^+ u: V5 m% @/ n9 _    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
5 q: ]7 n1 `1 M5 d7 T/ t# m* y]