实二次域(5/50)例2

lilianjie

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Q5:=QuadraticField(5) ;
Q5;
* X. [, g: f' S/ y6 H& tQ<w> :=PolynomialRing(Q5);Q;
* b! b' c7 o* Q4 t6 p+ {& a
( R2 X# }! w4 s# @: Y/ a) OEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
NumberField(M);
! S2 S0 M5 j5 t7 D2 iS1:=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);
Factorization(w^2-3);
  N. i6 a% w( h" a6 M* t8 i+ ?Discriminant(Q5) ;
FundamentalUnit(Q5) ;
& U6 L6 @; j; S) oFundamentalUnit(M);
7 N/ Q- o3 n3 W# r  ZConductor(Q5) ;
Name(Q5, 1);
& k9 ~1 e, A' ^& PName(M, 1);
. `( F$ B: |# v/ QConductor(M);
5 a4 S0 M# e2 R4 ~, uClassGroup(Q5) ;
1 ]) X! ~& ~0 @9 g! bClassGroup(M);
ClassNumber(Q5) ;
( f: K9 q' v; M( C4 `% M8 F& a( P7 eClassNumber(M) ;
/ I7 j% E/ B8 [" d4 X7 F
. N0 e2 D+ t$ @: \# oPicardGroup(M) ;
PicardNumber(M) ;
2 L- B* \0 _  d/ Q( M( s4 s; l
0 o+ ^. H# Q  F" ^7 `9 K
) P6 s/ s" d/ MQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
1 o  T; f  ]5 ]) _3 j0 m" B

- r2 [0 u8 o, K6 e- j6 E% }% dNormEquation(Q5, 5) ;
+ u) P9 l9 C& j  w" O0 @2 G* QNormEquation(M, 5) ;
. C. g5 z4 a& Y4 T# d1 H3 k
' E+ B9 Q5 s8 K- C5 h( b
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
# T. G" C: Q/ J& y5 M+ `4 \3 aUnivariate Polynomial Ring in w over Q5
7 h* E+ u' g6 u5 D8 XEquation Order of conductor 2 in Q5
8 j/ U* [/ h0 I" M$ @/ ]4 N6 BMaximal 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 $.1^2 - 5 over the Rational Field
- N! m" F! q) i) I$ e+ P5 @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
[
& f( z8 R5 \% C5 |    <w^2 - 3, 1>
2 j+ [/ b0 r* Z. S]
, {  w; o7 U) p& _5 B5
: a' [3 b! h1 e0 }5 [1/2*(-Q5.1 + 1)
-$.2 + 1
, J0 o( q" J* o3 p9 i7 T8 r5
Q5.1
* ?1 N# L' S; G( l5 T$.2
; n4 K! b1 H7 y3 g3 {, U8 Q0 \0 S1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
  b9 W9 x8 G) b5 EAbelian Group of order 1
" @/ i4 G$ q( u9 b( a3 LMapping from: Abelian Group of order 1 to Set of ideals of M
1
: |2 {0 N5 R! m& a+ a1
Abelian Group of order 1
( q$ x+ |/ M5 R: q: ]Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
2 J8 W5 q& A- @. Q4 T  Y0 o7 O1
$ k2 r( x0 |* y1 T$ G* nAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
7 S* x8 S7 P- I5 }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) ]
6 F& e, S2 _' o5 Ytrue [ -2*$.2 + 1 ]

" h# }: q. [5 w3 [3 L3 \' ]5 C8 O+ ?

* [' [2 y* S! [3 ^* _  F
1 D5 d$ C4 f+ r1 k$ _3 ]

- F! V9 h0 f% j" A3 H6 r

* Z; `# U% [( p2 I+ Y5 g


" H+ t* |9 K. D/ T==============
- S6 i3 N# ^/ F7 ]0 T" F
. x) p9 c7 i( w" O( k0 j+ EQ5:=QuadraticField(50) ;
3 J' q/ x. {& F, G* Q* CQ5;
. l, d( Q1 f( z7 g3 F: \8 w- W
" i8 k7 C' v$ T& ?$ ~& H! \  e8 ~Q<w> :=PolynomialRing(Q5);Q;
& d' S3 z% U4 LEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
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;
3 u$ X+ M7 k0 p% j8 v, yIsQuadratic(Q5);
, v: |& b; R" }$ ?( V+ {7 p% dIsQuadratic(S1);
& H- Q' Y6 h/ [; G" d. QIsQuadratic(S4);
3 \3 a& Z3 `/ [. V" ?. BIsQuadratic(S25);
IsQuadratic(S625888888);
; C  C6 ^5 U6 k  s: |7 u9 A) C1 g  lFactorization(w^2-50);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

% g- \: @: c7 a% j( eName(M, 50);
' {9 Q* I1 m+ J$ K$ P' {4 j! Z0 qConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
) V9 t, ]* C+ X2 O$ g" |# m& {0 SClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
2 r7 h0 x3 j* Q6 `  N1 E2 g
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
& k: B, {1 B) r3 [' Q' P" aNormEquation(Q5, 50) ;
NormEquation(M, 50) ;

6 x8 u; Z/ y; o- gQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
! z: e- A' X; Q/ e$ D1 S3 r5 Z3 yEquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
1 e2 A# p- F, T, q6 m6 Z, p$ \; `  gQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
7 u8 X- n* D5 j' eOrder 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
* ~& v6 q- ~- A0 J  b* i# mtrue Order of conductor 1 in Q5
$ o) b; D+ Q) s' A) f" y% Mtrue Order of conductor 1 in Q5
[
' x  _& m/ W8 d. @( z! ?    <w - 5*Q5.1, 1>,
5 x9 q  X3 E' h3 ?- j" w$ A    <w + 5*Q5.1, 1>
6 }+ r6 G$ {0 s: N4 O]
8
Q5.1 + 1
$.2 + 1
8

' C1 p( c; M; |& Z- o- _>> Name(M, 50);
       ^
  [( h, i$ j# sRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]

4 u9 h% S& c% L+ n. i) G. M1
Abelian Group of order 1
* `! y  N. |' R/ r2 B  SMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
' y! W% b4 x2 m- OMapping from: Abelian Group of order 1 to Set of ideals of M
: P2 p! f- A5 L5 e. |1
! @5 r" {1 ^+ X, \! @' ?: a1
Abelian Group of order 1
8 W/ ~9 w/ z* X/ B: SMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
Abelian Group of order 1
5 U: e' _: {3 e! D9 K: K9 WMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
Abelian Group of order 1
- \! x. ?; J) b( z" W6 u: fMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
true [ 5*Q5.1 + 10 ]
) y% N3 T3 s1 n) N# Wtrue [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

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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.

7 ]) v% O3 N/ t& ]
最小整解（±2,±1）                              最小整解（±7,±1）
                                                             ±7 MOD2=1
! @4 ?0 o3 b3 z, J; k
0 U# H7 ^$ `7 Y. \& x2 x两个基本单位：

lilianjie1

lilianjie 发表于 2012-1-4 18:31
6 ^* i3 i  Y0 p* T+ V) a5 @5 b基本单位fundamentalunit ：
5 mod4 =1                              50 mod 4=2

基本单位fundamentalunit

lilianjie1

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判别式计算Discriminant

4 _. Y7 f* t1 y1 W5 ^5MOD 4=1

（1+1）/2=1          （1-1）/2=0
5 v% [; P5 ]1 ?- g0 f- Y# u; K7 S
D=5
1 ~# c. B7 }' W( m! P

- }# }- a' Z, a. n50MOD 4=2
D=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44
' o/ C( q- l0 z7 w" Y) v( K
分圆多项式总是原多项式因子：
C:=CyclotomicField(5);C;
5 L, K( k1 K! _! ]& r. x: S3 {6 `CyclotomicPolynomial(5);

分圆域：
! S' e) {: K  K- c* k" [分圆域：123
7 z$ U' h4 J+ j# A: n
9 n1 ]7 I  ^; o( iR.<x> = Q[]
1 f4 N9 w3 w5 Z; j, KF8 = factor(x^8 - 1)
6 k! |) U4 _2 g3 Y, D* d. w# OF8
  l- N" J6 w# C/ b. V1 m, k! @
2 Y' _3 x8 V. @/ G! Y  w(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
# {9 `7 h: {# Q
Q<x> := QuadraticField(8);Q;
, S: {( @9 I# [$ d! U5 GC:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;

F := QuadraticField(8);
F;
D:=Factorization(FF) ;D;
2 C, b; C9 P" tQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
# c0 @0 s* t" S5 h" g9 U" pCyclotomic Field of order 8 and degree 4
$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
* x  n8 y' ?6 J# [+ D[
& |# G% _! |' Y% `    <$.1^4 + 1, 1>
7 h% k) P+ T9 l6 z) P/ ]0 I]

R.<x> = QQ[]
4 F6 k% }: V4 J. EF6 = factor(x^6 - 1)
F6
) ?) c: P7 ~- z) t6 U* @
+ ?& ?8 o, V4 R) q, B* L(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

0 }) t; b0 h  D8 S8 X1 x( \Q<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
5 N  W: ?+ g# d) JFF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
F;
" {# N" A  k5 h/ S% l6 x  kD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
$ N3 Q) b+ [8 r8 z[
    <$.1^2 - $.1 + 1, 1>
* G, D# ?; c* n, E]
) R$ u, h) i0 |/ ^) C8 m
R.<x> = QQ[]
- q0 z& Y; p* d& X; B7 GF5 = factor(x^10 - 1)
% A$ R5 m$ Q' k9 o! ~6 uF5
(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)
" O3 f% d0 V$ B5 M/ d- D8 |" x
Q<x> := QuadraticField(10);Q;
" p. ]1 }$ l/ v0 t2 f1 @C:=CyclotomicField(10);C;
9 ^- C0 i/ y7 P1 G1 LFF:=CyclotomicPolynomial(10);FF;

F := QuadraticField(10);
F;
7 x6 H1 V( b  J4 S6 C( VD:=Factorization(FF) ;D;
1 v( K1 f* v6 I& ]* EQuadratic 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
/ J( m1 j" J, x( s0 ^. xQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
3 L5 V0 I. O. ^; K& X[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
8 \/ O0 `- {0 M/ S5 r]