实二次域(5/50)例2

lilianjie

4 J! W0 q3 X* M, n# pQ5:=QuadraticField(5) ;
Q5;
/ E3 }9 _' r3 l" B1 A% U  |( G) ZQ<w> :=PolynomialRing(Q5);Q;

8 y0 T* K( I7 ]9 w) |EquationOrder(Q5);
. N& [1 d( N& S! JM:=MaximalOrder(Q5) ;
M;
* g( z7 h& a( H5 Q( bNumberField(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);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);
Factorization(w^2-3);
Discriminant(Q5) ;
" Y5 G- p- G$ _+ tFundamentalUnit(Q5) ;
: i/ ~; u) t9 Z  S" XFundamentalUnit(M);
" h2 i3 `6 J# F7 a7 {Conductor(Q5) ;
0 Y5 U# G" Y$ @, j& o6 g8 I, L+ YName(Q5, 1);
Name(M, 1);
) S& c7 `% Z2 l; p( f) SConductor(M);
6 b9 z  {0 B, ]  [. V( mClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
3 H/ m; y" r5 ]) B9 K- O
5 _, t2 y' ]" R) H2 p' M5 U; ePicardGroup(M) ;
5 {# F+ W1 r) C$ y) q  k4 x3 APicardNumber(M) ;
  t! n( g% d3 ]! C

QuadraticClassGroupTwoPart(Q5);
, M* m4 x. Q  r$ l9 a% SQuadraticClassGroupTwoPart(M);


NormEquation(Q5, 5) ;
NormEquation(M, 5) ;


7 [# Y3 v1 N5 B4 U) S6 b; zQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
0 L: o- i" D7 y0 C1 [# aUnivariate Polynomial Ring in w over Q5
' G5 H( z  P" }3 O# fEquation Order of conductor 2 in Q5
! K* |& D' K& k8 D) p" d& h$ L6 M. qMaximal Order of Q5
# n) {$ t: l3 R( a" V6 r- gQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
1 K' t' O% x5 GOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
8 A; E; Q0 u4 i1 H6 ctrue Maximal Order of Q5
true Order of conductor 16 in Q5
7 i2 Z: |- c% @; B1 d* Htrue Order of conductor 625 in Q5
! Z- t/ }! N7 f$ J, R% Itrue Order of conductor 391736900121876544 in Q5
[
3 g$ H/ A5 A0 M5 \8 Q1 O    <w^2 - 3, 1>
]
4 \: P! i4 P2 Z' F! z5 M; n5
1/2*(-Q5.1 + 1)
' q) L+ l( D% U0 ^  \8 O-$.2 + 1
5
Q5.1
$.2
: B$ y4 |& u7 U: O1
" F9 b! A' V+ p* S: I. CAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
2 a) A5 u3 i! ~Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
4 g* ^( ~) Z/ M; }1
3 E+ S  e" H7 m: L% TAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
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]
3 b' a8 [) m) Z; r& h( c$ BAbelian Group of order 1
: S. U1 P7 W$ D4 o) s, EMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
& V! z' m) Z1 ]! n9 b6 O& H5 given by a rule [no inverse]
; p2 q! v4 d, M9 t- Ptrue [ 1/2*(Q5.1 + 5) ]
true [ -2*$.2 + 1 ]
7 P0 x; {0 d, q+ M7 H' f8 B# B7 W% L
8 Z4 P" ~, G3 F, r6 E. N$ F( h. S

" ^9 }/ f) ^3 \# K, ^


- o& h0 n% i; _) B9 Z- d

  l. X% D2 f9 |6 u4 b7 v
& ~3 N3 V4 ]7 b% o! J: T7 ]

==============

Q5:=QuadraticField(50) ;
0 B1 @0 u1 i4 A; GQ5;
$ g) i' V1 g9 C
Q<w> :=PolynomialRing(Q5);Q;
$ B# ~3 y3 n4 \. oEquationOrder(Q5);
1 N1 m0 K7 h# `# E2 U+ H7 t  EM:=MaximalOrder(Q5) ;
2 s: O# A! p1 M% ]3 u- c, w* oM;
* Q: f+ M7 V7 P/ ?2 UNumberField(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);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
6 v  e1 b/ G, X* p2 P* d# MFactorization(w^2-50);  
3 D% D3 I2 F2 ]6 {8 h' WDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
+ l% u, R. O2 [1 A9 rFundamentalUnit(M);
5 y2 `8 k4 L4 Z" i+ ?$ QConductor(Q5) ;
7 z7 q. F, q+ `$ q* p9 b+ b$ y
Name(M, 50);
Conductor(M);
: T6 O# v+ J0 ~" _! ?! v/ B& TClassGroup(Q5) ;
8 F8 L! Z( `( e( |( j- E0 z0 pClassGroup(M);
5 p/ M  D9 S9 v5 H1 B  QClassNumber(Q5) ;
$ J0 q: e- h% `& B7 {ClassNumber(M) ;
1 P- r6 W! L) U9 }9 f; uPicardGroup(M) ;
' C" p; `8 x$ k" A4 gPicardNumber(M) ;
6 R1 C* P4 h4 A, R, d& P% w( z
9 J  G5 w  K7 d/ I0 YQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
( N2 E! E7 ^1 S$ R+ hNormEquation(Q5, 50) ;
NormEquation(M, 50) ;
2 X7 V; j8 @* l8 V/ p8 K
2 R' Z3 B( y6 L$ H: |Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
& _4 R% v4 j+ I) T' WUnivariate Polynomial Ring in w over Q5
/ L3 \  `( n( ?9 G& ^* CEquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
3 u" D8 i! Z3 X- W' J. MOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
true Maximal Equation Order of Q5
: \$ v- a9 l: btrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
" @, |- @6 ]1 m! y3 |  ]- {[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
7 {4 \& P6 |* X% L2 N3 g8
6 m3 }- m% a; ^Q5.1 + 1
: D! X8 y( y5 J0 g; Q, }' ]8 C! O; O$.2 + 1
, d8 d3 g/ Q- E+ w- z/ ]8
" b5 G) M1 Y% h3 S& v
>> Name(M, 50);
       ^
Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
: u# L) b, N4 K2 U4 x& x3 H
" V  K3 B) @2 v9 ]1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
4 L" b8 Y8 [2 G/ H2 [Abelian 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]
0 s2 D' `# Q& _8 c1 y/ }1
  t1 C9 g9 k+ M# P0 ], c' `8 |/ kAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
" {6 Z2 v0 r& |5 i/ r, y8 given by a rule [no inverse]
1 w6 }% e: Y5 r6 E% ]/ strue [ 5*Q5.1 + 10 ]
$ c+ A  z5 V5 `, [7 Q1 D2 G9 r9 C3 Otrue [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

, b0 q. Z! o: u; i1 c9 E9 C  _
! ]# ~2 C  J' ]; F" c基本单位计算fundamentalunit ：
5 mod4 =1                                              50 mod 4=2
4 j9 m+ g& E! r7 @* s  I* Q
: d1 r1 Y; G* d2 r# X+ Z6 Y x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
, w* l) Q: X/ h& |, W& E; _0 w! k  G x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.


  H& x* B8 c' N" [最小整解（±2,±1）                              最小整解（±7,±1）
" g( J, `) R' _; H5 D- z: J  x! M                                                             ±7 MOD2=1

! A) d! h! D: c: _( ?1 k! }两个基本单位：

lilianjie1

lilianjie 发表于 2012-1-4 18:31
/ X) K4 q0 f, M. M基本单位fundamentalunit ：
5 mod4 =1                              50 mod 4=2

8 M) y3 g* J! _2 A基本单位fundamentalunit

lilianjie1

8 K4 e4 |" A1 X. l  a. T& Y& Y判别式计算Discriminant

5MOD 4=1

（1+1）/2=1          （1-1）/2=0
/ x8 d: m; o  T6 n% y3 Z+ g& Z& b
5 U/ i' H3 F1 S9 UD=5


50MOD 4=2
- D' R$ s0 n. R9 d, _D=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44
9 s: o4 B9 j7 e( Q! l; v; P; E+ M
分圆多项式总是原多项式因子：
: T& g% F; y, M7 d! K, T% MC:=CyclotomicField(5);C;
, }2 ~( U% F+ V0 a6 X- c7 `CyclotomicPolynomial(5);

2 ^; h+ T+ Q) d0 Z% n; n分圆域：
  O  i, f& V* ?  D3 G分圆域：123

. v% u, u) k5 ?) J& b# ]# a6 PR.<x> = Q[]
F8 = factor(x^8 - 1)
. C  E  L2 g- ^F8
( l7 h- B- g3 |( J! E6 Q
2 I# U) F5 {! f0 ~5 u4 u(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;
FF:=CyclotomicPolynomial(8);FF;
7 K" m! w) a/ r: V1 u3 k1 _
F := QuadraticField(8);
8 ^3 r5 j4 r5 v) _0 }F;
& n& m; q, G) g0 ?: [D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
+ Z1 v* p5 O; f9 z: f! m/ OCyclotomic Field of order 8 and degree 4
7 t; i& u5 V1 |* s; z0 T$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
    <$.1^4 + 1, 1>
]

3 y" W" V( U3 S& T9 B, F, b$ }R.<x> = QQ[]
4 m5 s, ?# r5 u: D1 d6 qF6 = factor(x^6 - 1)
1 h# b( w& ?: H2 ^+ b/ R5 dF6

* M3 C, e2 C6 K- F" U: N1 _* H(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 _: y7 T4 E( l( u! ~9 n* NQ<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;
& g2 v, m( ?, N6 [3 O' g# ?4 B, Y" t% z0 |
F := QuadraticField(6);
  H4 B( J. I; g5 v1 TF;
- W8 s% w, w, b, _% |D:=Factorization(FF) ;D;
# D0 T3 b& F+ J6 XQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
( n, F& Y7 K( J8 O$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
- W, g  `  b3 q0 x# W3 V2 A$ z0 x[
/ @7 n3 n3 p$ I$ `, z' ]- I9 p    <$.1^2 - $.1 + 1, 1>
6 d$ [3 Q7 j% X3 x# i: B& R]
: k3 g! Y/ [( R  u" K( B
# Z; f: l  O* n: G( M; b2 S, XR.<x> = QQ[]
& Z7 y: n3 t' f$ FF5 = factor(x^10 - 1)
F5
' z$ Z% g* A7 g( E" I" W1 B: ~(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
% [/ G5 e! V, X! O1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

# k: }  R: ~0 R( ?- LQ<x> := QuadraticField(10);Q;
1 V& n4 A/ A- ]1 W; l. T; _C:=CyclotomicField(10);C;
) j4 n# s' D0 R# o, Z/ EFF:=CyclotomicPolynomial(10);FF;

3 C* ^+ m4 H. ]3 c2 J8 m- d9 l' SF := QuadraticField(10);
3 j5 p) ^7 `  fF;
D:=Factorization(FF) ;D;
Quadratic 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
3 `, d" w& I/ `Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
8 ~3 F& p/ y5 G  _/ B  q7 b]