实二次域(5/50)例2

lilianjie

" d  @# z: C! M& V% ?
Q5:=QuadraticField(5) ;
Q5;
1 ^; A$ ^9 o: i* RQ<w> :=PolynomialRing(Q5);Q;

( K& m) \9 \% V6 N, DEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
8 G- o0 q+ L* M, X; L8 lNumberField(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;
8 V( t2 H7 j9 h4 \+ Y1 Q6 l0 qIsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);
Factorization(w^2-3);
2 ]; J8 e# y9 y4 o; j+ ?Discriminant(Q5) ;
) v+ N/ D; R: g% Q. T( B  }FundamentalUnit(Q5) ;
$ f* x9 a! D; N9 I7 v1 KFundamentalUnit(M);
; s" \$ r" S/ a: C6 KConductor(Q5) ;
Name(Q5, 1);
Name(M, 1);
. V" y' ?& n4 X& W$ x. l/ HConductor(M);
ClassGroup(Q5) ;
( J5 r1 j; r8 ~! a% c9 ^7 GClassGroup(M);
8 o6 X) ?, `9 V& u3 YClassNumber(Q5) ;
ClassNumber(M) ;
% J2 R6 h4 A/ A. t+ [" c& f9 O3 O# D
- w6 V# t  t, z" Q+ w% [$ WPicardGroup(M) ;
' V( @9 c: b* k7 E' q6 }" G2 dPicardNumber(M) ;
$ \+ _# x4 R6 C! b* C
4 J% B4 g/ a# |5 D5 F- t
4 f; o* q. K' U1 Y/ aQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
& B. j! m$ ^0 D$ n+ H0 t4 ~

, `+ Y8 K0 \* `7 v! BNormEquation(Q5, 5) ;
8 d8 U' `7 H! {  ?7 c* B, r4 f2 A: lNormEquation(M, 5) ;

; o7 D2 t$ e/ [5 Z3 O: H, [
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
" @2 f) x0 t6 C! DEquation Order of conductor 2 in Q5
, q9 m" S9 U& E' n8 Y0 q( L0 fMaximal Order of Q5
( w( M& j3 K+ @4 E' F9 Y& c) @- b* xQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Order of conductor 625888888 in Q5
. s: n. W2 _8 @" F3 s3 S2 rtrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
- U7 t$ o/ h- x% S8 Ztrue Maximal Order of Q5
  x* ]' _9 [" H/ m( I7 [9 Ttrue Order of conductor 16 in Q5
true Order of conductor 625 in Q5
1 G4 P$ j3 ?/ H+ t5 m4 jtrue Order of conductor 391736900121876544 in Q5
) q# d% [- b5 h3 q* a[
  p2 o8 ]; N! q! U1 H( N    <w^2 - 3, 1>
3 M/ K7 m! d: a; h3 {3 V; C]
5
6 m" N1 ]& _/ F/ [+ U4 L1/2*(-Q5.1 + 1)
0 {: h  [, ^2 {9 S% a8 I8 \-$.2 + 1
2 E5 R2 C8 @' P# Z: r6 d5
Q5.1
6 S  i: {' }# x/ U  e: s5 s  ~$.2
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
' [9 l' T9 {! T' ?7 m" U/ b6 ZAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
: k! ?. g" U8 z0 `  h1
8 D$ H' L1 q! P% ?0 O; W% G0 cAbelian Group of order 1
Mapping 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) ]
true [ -2*$.2 + 1 ]

! G& z3 j5 n& Z& u  l


$ E# M+ o, |/ D6 ^$ r
  j  y. Q7 J2 x2 t* w


! [" H1 I4 L% d7 p* p- H. k


==============
" p' A) B" e7 J* f' G2 G7 L& _  _; o
Q5:=QuadraticField(50) ;
Q5;
  u/ J9 o  S' C$ H0 X
- \( v: K$ f& v2 |) \1 Y. _: OQ<w> :=PolynomialRing(Q5);Q;
: r. g( g0 q! l' s' p* ]$ 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;
IsQuadratic(Q5);
6 Z" ^/ z5 ^0 W- U) d+ gIsQuadratic(S1);
IsQuadratic(S4);
0 ~7 l, D$ J' ^1 d% KIsQuadratic(S25);
IsQuadratic(S625888888);
6 t3 |7 |: \$ L2 }1 |) dFactorization(w^2-50);  
' S) ~% u: A5 I0 pDiscriminant(Q5) ;
7 o9 X: {0 u! ]5 X; HFundamentalUnit(Q5) ;
  T1 ]3 r: K7 _/ [5 xFundamentalUnit(M);
Conductor(Q5) ;

Name(M, 50);
Conductor(M);
. D5 n) j: x+ g' W5 lClassGroup(Q5) ;
! \$ n4 q* H" Q* zClassGroup(M);
ClassNumber(Q5) ;
  ~& g4 [1 L+ `6 _6 ?7 vClassNumber(M) ;
PicardGroup(M) ;
+ ?3 E! N+ F" a% ~PicardNumber(M) ;
" v8 P' y& V8 {) p
6 r4 p/ N/ j6 fQuadraticClassGroupTwoPart(Q5);
' o2 a5 W6 W" ZQuadraticClassGroupTwoPart(M);
2 L9 w/ q, V3 B. [5 R2 o: r" lNormEquation(Q5, 50) ;
NormEquation(M, 50) ;

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
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Order of conductor 625888888 in Q5
! D! f, i1 f7 x. ptrue Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
; ?  V9 z% I& K& E1 _4 utrue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
! C& H3 l5 y! f; ~$ A2 _true Order of conductor 1 in Q5
' N4 ~! M  {7 `2 }& _* @( itrue Order of conductor 1 in Q5
8 v$ R. }! B, @2 [% m0 a# S[
    <w - 5*Q5.1, 1>,
# q2 R  l  \7 @; z. D3 Y5 W    <w + 5*Q5.1, 1>
2 l  K' R  D  g- I! @+ {]
2 W1 g2 D5 `+ K( o: M: j8
# a/ d* P7 `& K$ W- gQ5.1 + 1
  L- G$ f" s" A& r1 I$ z, C$.2 + 1
" E/ B5 L8 i( J9 E+ a3 T8
+ B2 o* R6 |1 R# M2 x
>> Name(M, 50);
) S; X  k8 |6 g0 O8 }# E* c/ a4 |       ^
Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]

1
Abelian Group of order 1
) ?6 h2 p# Y; E0 B  \. N- EMapping from: Abelian Group of order 1 to Set of ideals of M
7 i+ `- B4 H8 R' w- f. {( T& oAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
+ T. O: H& k9 X" c; N; y1
6 [0 W+ C( e- G; n1 h; lAbelian Group of order 1
" E3 T1 W3 a2 n  zMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
Abelian Group of order 1
' N& j  t3 U$ H. RMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
% n- [1 f7 H; b  P4 s8 K8 given by a rule [no inverse]
true [ 5*Q5.1 + 10 ]
) g$ ^) O0 O6 B5 S+ utrue [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

+ w- Y1 q" w- ]. x- L
$ o; m9 ~; N! D0 y" ?* o6 ?基本单位计算fundamentalunit ：
: K2 }- E- r) t" X5 mod4 =1                                              50 mod 4=2

x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
; I' L3 g  S( y x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.

" Y9 I8 Q4 X3 k
7 v' ]7 k' @2 Q4 B% [1 p# l9 }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
基本单位fundamentalunit ：
# Z& D8 R1 ]. `6 j$ @5 mod4 =1                              50 mod 4=2

基本单位fundamentalunit

lilianjie1

* v1 z! w( }! C6 _
! y, Y" M' m. u( y判别式计算Discriminant
' s! [" j( H, F: e. `
5MOD 4=1

（1+1）/2=1          （1-1）/2=0

" R2 d# c+ R: B. G2 K& b+ lD=5
9 l- x* S- O, h' l, y: g! Z
! [" p5 p& V$ d- F$ J- J
50MOD 4=2
D=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44

分圆多项式总是原多项式因子：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

* L7 f7 [' h% e9 d) H( Z# x分圆域：
* b1 M9 v4 C. h' y. X/ J, y. M7 p, E分圆域：123

R.<x> = Q[]
8 F7 U" Z; V6 s9 TF8 = factor(x^8 - 1)
F8
( p5 A. \/ O: S# ]# d( W
1 r2 ], l1 U; L# |6 H; X(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

# ]* b1 d. r# b" w  o, q% gQ<x> := QuadraticField(8);Q;
4 E$ K* D( H( Q% u3 ?. dC:=CyclotomicField(8);C;
, {: v0 c2 Y' V" u, UFF:=CyclotomicPolynomial(8);FF;

/ R7 n: }4 w" I9 ^F := QuadraticField(8);
F;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
$.1^4 + 1
3 X  S! {$ i2 Y  F+ T, ]- ^# dQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
  S3 y7 j( ~5 ?6 i[
$ k) ]( N9 P) M3 L8 v; A, s    <$.1^4 + 1, 1>
. l: G" c  ?8 j4 s]

# V) f1 B+ `& |2 E2 oR.<x> = QQ[]
$ O0 G+ k4 F( C9 SF6 = factor(x^6 - 1)
F6
7 G- s, ?  t; j' ~& c
7 Y( k$ v8 J" p; D$ p- U( ](x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
; l# V. F. L2 Y8 s0 @( g) V5 W; v
Q<x> := QuadraticField(6);Q;
' ]. C* y/ @) h/ aC:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;
; F, k1 Z9 W: J* m( h8 J) _8 j
F := QuadraticField(6);
F;
" e/ ?% l, _: H. a. W2 mD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
- f3 E5 j; Q7 N6 R: yCyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
    <$.1^2 - $.1 + 1, 1>
5 @, }4 d3 l( W( Y]
7 x3 z! E( E$ g9 y, O' V) q
R.<x> = QQ[]
2 a( B3 ^/ d" F4 w% ?8 X$ sF5 = factor(x^10 - 1)
" R8 }8 j( @7 ^& t" w; I- `; LF5
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
. W5 M' r9 A$ ^7 R1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
, [8 S0 ^  r8 S& T! v9 j+ Y
Q<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;
; b6 A9 v5 C2 t# Y
4 G6 ^5 w6 C/ |' q! h% H9 AF := QuadraticField(10);
5 e4 Q0 v& s$ y, IF;
D:=Factorization(FF) ;D;
8 U4 P' o* v5 {+ i: fQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
9 N; P1 z: n! |  g: |' f* {- lCyclotomic Field of order 10 and degree 4
0 o) m% g! P. G2 a% t5 W$ S0 U$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
! \: v. |' ^+ [3 W( N$ [[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]