实二次域(5/50)例2

lilianjie

9 X0 C% b9 A: N5 A7 t: I0 u9 oQ5:=QuadraticField(5) ;
Q5;
Q<w> :=PolynomialRing(Q5);Q;

# `8 D* E- O7 S2 dEquationOrder(Q5);
3 w) U( ~+ L, ?M:=MaximalOrder(Q5) ;
2 y, ]5 O- L5 [  t9 i( \5 H# hM;
3 s, U6 U* o/ J1 x1 r6 b: hNumberField(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;
/ a& e, p0 ~9 qIsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);
Factorization(w^2-3);
& v5 Y: ~6 }; a; t4 ^9 QDiscriminant(Q5) ;
# |! k* }1 I9 t# Z5 ^' @' d. a( [FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
9 \) W: I8 H' a6 k  |" T$ H+ a+ x. |Name(Q5, 1);
Name(M, 1);
" U  Z1 e! g' b% \$ {) [, o7 z! JConductor(M);
ClassGroup(Q5) ;
2 U3 y+ y! ~$ |7 `/ l- n: A" S1 nClassGroup(M);
ClassNumber(Q5) ;
) v4 D1 G7 R! |$ i: f2 AClassNumber(M) ;

2 S- g1 \: A4 q( ?, \! gPicardGroup(M) ;
PicardNumber(M) ;


  N4 V: _9 _" b9 ]. M" OQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
9 @! z! Q, O* ^5 I. U6 ?. W

NormEquation(Q5, 5) ;
; @- n7 V3 K- ]NormEquation(M, 5) ;
! j6 O; d' e: k, p4 t

Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
% k  d) r; N/ W7 r1 oUnivariate Polynomial Ring in w over Q5
1 ?4 |3 k3 ?. J0 J6 C) j; REquation 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
" Y. M9 ^! U' _* A8 a- etrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
; k* f6 B/ G0 z& ^: Q  H1 U# w& Utrue Order of conductor 16 in Q5
true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
$ ]: f8 v/ h3 v( w[
    <w^2 - 3, 1>
( Q0 Z8 q. m9 G  _]
5
+ I8 n* E1 m& g) f1/2*(-Q5.1 + 1)
-$.2 + 1
5
Q5.1
$.2
/ L% Q( s0 s3 ~+ r: A4 o: u1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
0 n& \1 C* `- l1 L, c+ D% oAbelian Group of order 1
1 M* p1 p! M* `( N6 c& m& g# `+ u( W% ?Mapping from: Abelian Group of order 1 to Set of ideals of M
1
+ s+ W2 q+ j. j/ }, \1
1 Z! G& }7 d) k6 H" sAbelian Group of order 1
- S, ]- h( Q) a6 O3 AMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
- T, N% ^0 l3 |* m. P1 k; D/ c2 ginverse]
1
Abelian Group of order 1
" x: V) ?9 }7 H6 Y1 `! Q# jMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
true [ 1/2*(Q5.1 + 5) ]
& e1 q1 }1 w8 O$ _1 i% ctrue [ -2*$.2 + 1 ]

1 y- b) n; [% _# O/ A5 a! }/ R1 B
1 H8 k: j, C3 ?/ _. e) k; Q* R
5 j  @3 ]- C# S) P9 ?0 x& t! C# }

# [0 j. j: J) Y% `; l
1 Y8 g2 v( T; [3 V
1 |5 P  v6 d& [# m+ B# R7 M9 Y


& u/ r+ U, s% I$ G4 j
0 B! W- @% ?$ e==============
+ _; |8 ?/ O+ Z
Q5:=QuadraticField(50) ;
" C% r: s5 M  l* ?) |- R/ |Q5;

* m& C0 Y! M  d# OQ<w> :=PolynomialRing(Q5);Q;
. M( Y+ W( i& v- SEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
6 E: y7 t- M* ]M;
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);
IsQuadratic(S1);
0 F) E4 a- V- KIsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
& }5 B3 h# Z7 k# LFactorization(w^2-50);  
3 A' O( N6 y. r0 C3 \, zDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
! C( @( d! y( yFundamentalUnit(M);
. V7 E) N1 s" O5 \: y) {Conductor(Q5) ;

; p* @6 p% }5 ]+ s+ V' bName(M, 50);
Conductor(M);
# [2 S% _! |, H  g9 KClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
. U! t9 ~  ]7 T3 gClassNumber(M) ;
2 c8 h, h  B$ D2 y7 kPicardGroup(M) ;
$ O, W0 I7 r1 b2 @7 H/ P6 sPicardNumber(M) ;
' J& M$ O4 Y" `2 ~# R
8 Q, x5 H7 m6 Z+ R: uQuadraticClassGroupTwoPart(Q5);
% q4 r$ a  y* @& m* wQuadraticClassGroupTwoPart(M);
NormEquation(Q5, 50) ;
NormEquation(M, 50) ;
( Q8 e" Z( E! @$ S% L% M1 F
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
9 T$ R3 T( g8 b  zUnivariate Polynomial Ring in w over Q5
" i+ ~/ @  M2 i' KEquation Order of conductor 1 in Q5
" J# t) ]- K) p8 EMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
true Maximal Equation Order of Q5
9 h" w3 X3 \& strue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
7 C+ [! J0 M* q1 V& C4 a8 D* F/ q[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
8
Q5.1 + 1
  U, o8 K0 q# e6 P& R& E0 }$.2 + 1
( A4 N4 z! U- v$ N* Y+ ^8
; w: l8 G. a4 O, a' A8 M3 e
>> Name(M, 50);
* Z8 F/ J! @& e1 ^, g       ^
! M. F7 m1 o" k/ B; DRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
2 R% X& u! d- o8 J
& G9 j" W% ?2 ^1 {; m+ ]& [# ~8 t" P( I1
5 h' k: G( m8 j3 `Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1 ^- N5 P/ ?/ D- [1 h9 AAbelian Group of order 1
; e- e4 w# x- v+ _& L/ Z; {Mapping from: Abelian Group of order 1 to Set of ideals of M
1
1
Abelian Group of order 1
  \& x- ^. p/ ~7 t# M6 `1 _Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
6 R& Q6 t0 D8 z3 J: Cinverse]
1
, O+ L6 Q5 ]0 P4 ~" E( dAbelian Group of order 1
2 _; [1 s0 M( _) a$ dMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
Abelian Group of order 1
& H% y& ^8 m# t5 F2 ?7 [* R2 JMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
9 x! h- U1 F+ I  Ltrue [ 5*Q5.1 + 10 ]
true [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

8 O4 U" u$ M* |  N- S6 [+ v3 L基本单位计算fundamentalunit ：
7 N- A9 P! I) u6 }, ~8 H: O0 c5 mod4 =1                                              50 mod 4=2
0 D1 k( v5 b2 d
* o9 b9 Q- T) V x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
% q( G1 \& X3 F3 Q* D x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.
+ f' N7 C! X! a9 P' e& k

3 g0 d/ o3 c0 @+ n( C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
: b2 k4 S, B( i, s0 F8 t+ D/ R基本单位fundamentalunit ：
5 mod4 =1                              50 mod 4=2

# j) v$ y; n5 z* d0 d基本单位fundamentalunit

lilianjie1

/ D1 i5 ~8 |8 l. j( g
/ A# \6 i- M) k( r# Y/ [判别式计算Discriminant

2 d) d9 U2 z. Z8 [! E4 k5MOD 4=1

1 N' X. {/ w4 }6 @: a6 ^8 c  n（1+1）/2=1          （1-1）/2=0

D=5


$ K: E8 i& F5 q  g2 J  a) |: u2 _50MOD 4=2
D=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44
7 G$ D- j( Z% z0 t5 z0 x+ E
0 \& K. |! m, j/ }: b6 Q9 B* ?1 v  O分圆多项式总是原多项式因子：
C:=CyclotomicField(5);C;
5 h6 g2 `8 l, ]; P, N5 @2 `CyclotomicPolynomial(5);

3 D5 r9 e  x% S" \分圆域：
分圆域：123
& Z1 ]" b. {5 v8 b
0 N" J' G2 J' \% w$ i7 P2 a8 h- bR.<x> = Q[]
F8 = factor(x^8 - 1)
( I3 u5 K: t) k$ D9 ~4 s/ t% b/ HF8

(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
* A: o7 c  x6 c+ v
Q<x> := QuadraticField(8);Q;
& M7 v# E% Y6 E$ |$ K1 h. }C:=CyclotomicField(8);C;
* ]& H2 u) _3 S9 t* t, m$ kFF:=CyclotomicPolynomial(8);FF;

F := QuadraticField(8);
F;
7 l+ k9 ]$ J( O- M: x" [: [+ ^D:=Factorization(FF) ;D;
0 u8 M7 `* }4 C& }9 XQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
9 m" D4 V1 Q* Z1 l2 Z2 B- \$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
% E/ L5 t  }, Z7 X5 S[
, B* S4 c# I/ P( v% H+ ^    <$.1^4 + 1, 1>
( L9 S; n+ h; [1 x( R) O$ R]

0 `3 w0 J5 _: NR.<x> = QQ[]
2 ?/ t' e& k. PF6 = factor(x^6 - 1)
F6

7 G' s# `4 R) \: [1 Q- Z0 ?(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;
7 _7 j% P' y, B/ Q; EC:=CyclotomicField(6);C;
8 Q6 r4 V1 T1 |) d6 {FF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
F;
% Q$ O% q0 b8 y+ s0 WD:=Factorization(FF) ;D;
4 a4 g' j7 A" S) O. w# A9 x& y- NQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
0 {0 R6 [# m* X$ i5 I% N4 A' bCyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
* U' D% z1 D/ W/ RQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
2 F4 c  s4 N7 @+ b[
' C$ D$ [4 U, N    <$.1^2 - $.1 + 1, 1>
0 g0 g4 O4 m. @: y]
4 ^( Y2 N/ L1 o$ i+ P0 }' D
7 z- c0 r  a0 |/ l) L9 Z, j7 zR.<x> = QQ[]
9 c! E6 L+ }$ SF5 = factor(x^10 - 1)
2 y  u( r4 g2 q4 {5 k, Z1 yF5
(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)
" t! q6 D1 t: K* o2 Q# {
Q<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

  H: o$ ?+ B/ }. v+ }0 [F := QuadraticField(10);
. [; j9 }% h! t7 cF;
D:=Factorization(FF) ;D;
5 X9 P2 Z5 {' j$ B' ^# @& C' xQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
" d& \1 q: o- l" @# ]$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
4 M. m- l; H  f2 v& W; s+ {9 ~[
, K. w" n: T0 W/ j    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]