实二次域(5/50)例2

lilianjie

( J9 y$ G+ L1 Z. y& P6 LQ5:=QuadraticField(5) ;
. u9 `- F5 k- D9 f* Q6 S$ H$ o( qQ5;
# \% w  o  Z8 S  v4 D# W, ?8 ?( VQ<w> :=PolynomialRing(Q5);Q;
: Z6 U3 h8 r" w. x7 O5 K1 t
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
, G, Y: q5 {# n+ n! D9 zM;
NumberField(M);
9 M$ l% B" Z# j1 y/ AS1:=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);
8 u/ M% H) \0 m$ F3 @% hFactorization(w^2-3);
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
' S" [$ a. B, j1 o2 R2 Q* n7 nFundamentalUnit(M);
Conductor(Q5) ;
Name(Q5, 1);
Name(M, 1);
Conductor(M);
' J- {; e# _0 a: U% s, e6 fClassGroup(Q5) ;
ClassGroup(M);
7 l6 v" O/ l/ y: a0 M/ T. J- n- ?ClassNumber(Q5) ;
% K6 G7 b9 H2 V9 c6 s, X% ZClassNumber(M) ;

PicardGroup(M) ;
6 x2 Y: j0 P+ b  v4 PPicardNumber(M) ;
* G* H1 e' F) C9 a. g

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
, s$ q1 F  q! C
+ t/ w5 R! J1 f9 r9 Q1 _; Z
NormEquation(Q5, 5) ;
NormEquation(M, 5) ;
% U1 M7 e9 k# g3 @
% Z. b5 H+ P% P
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
1 s# G5 E; ]: k/ CUnivariate Polynomial Ring in w over Q5
- A) p1 c' A& G7 k% xEquation Order of conductor 2 in Q5
Maximal Order of Q5
# f" V5 g2 R4 o4 D3 fQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Order of conductor 625888888 in Q5
3 a1 N- q  f0 S9 q3 }! s, M, Y# gtrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
# ~1 U7 R5 p* _9 a$ o+ r& ~/ h0 Ptrue Order of conductor 16 in Q5
- M; ]6 [4 A. |5 V6 E8 r; X( Itrue Order of conductor 625 in Q5
% _3 L! Y6 u  J+ U6 S0 I% p  O0 H/ Vtrue Order of conductor 391736900121876544 in Q5
+ P2 g4 |( o* T8 j/ m6 K& \/ ~3 o[
7 Y5 M/ m7 C: T    <w^2 - 3, 1>
]
7 u) w! L6 F, T9 I5
* A6 W+ A( B1 G' ]6 c8 A1/2*(-Q5.1 + 1)
-$.2 + 1
" X" B% t; g5 g- z5 \" s5
Q5.1
3 m+ r0 \' a0 j6 ^" F$.2
, \1 t) e7 O4 l/ H3 H1
Abelian Group of order 1
7 h+ c; r: B* ^0 GMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
: B9 E5 ]! F" L+ D; M2 [Mapping from: Abelian Group of order 1 to Set of ideals of M
1
$ Z$ a" X, J6 F1
Abelian 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]
7 ^  S0 V9 ?* n% Q2 S4 Z& 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]
7 Y9 j) L* [- r: Strue [ 1/2*(Q5.1 + 5) ]
true [ -2*$.2 + 1 ]


) v9 J* L# I8 k6 O9 [

- E- u& ~9 R3 k# m) [0 a" ]2 F; v


% t8 p& R2 ^  E


' a" Q5 a" E6 `/ X
==============

8 C9 J. B8 ~1 c! lQ5:=QuadraticField(50) ;
  v/ k2 H( o- n; Y4 e+ {* ^Q5;

; ~/ ]6 u$ ^" Z, l) M; K, pQ<w> :=PolynomialRing(Q5);Q;
2 u3 ^5 N7 R% d' m; |2 t9 rEquationOrder(Q5);
3 n% l2 J9 j0 z/ vM:=MaximalOrder(Q5) ;
M;
NumberField(M);
2 M% l" w3 Y% t8 h3 vS1:=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);
5 ?5 h% n9 j/ _: \; E  w: Z5 lIsQuadratic(S1);
IsQuadratic(S4);
# w+ T' |0 n6 K+ }' xIsQuadratic(S25);
3 W4 {8 X' F" w- J1 ]  N. }IsQuadratic(S625888888);
; h( z, F& |2 l: R0 e# X. P- l5 ^Factorization(w^2-50);  
  g' \- P8 A3 ^Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
5 c4 D; F6 ]: p% o( |/ x
/ \( P, h/ g6 s- y4 uName(M, 50);
7 M: X7 Q8 i3 [; q# uConductor(M);
- E8 }7 S2 N/ YClassGroup(Q5) ;
$ J8 x. x* Q& n$ v( I' GClassGroup(M);
$ b) F" Q6 W. I! z% fClassNumber(Q5) ;
ClassNumber(M) ;
5 Q5 _; b' q& ?. i# bPicardGroup(M) ;
PicardNumber(M) ;

/ h! i3 p4 ^, S- s# FQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
: n: E: g7 P6 B" {; p) TNormEquation(Q5, 50) ;
NormEquation(M, 50) ;

- F; Y+ }1 n( \; H* XQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
$ `) O0 o& L# J$ V7 t( _Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
* Z% n- W; G; t1 L1 dOrder of conductor 625888888 in Q5
: E" q7 Y* c$ xtrue Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
  _: [  v: j; ]' k$ x! }; M+ _true Maximal Equation Order of Q5
7 O5 C! k6 ]0 d5 s6 o$ R; [true Order of conductor 1 in Q5
! Z$ M7 e' l/ W4 t8 v- J6 Q% Dtrue Order of conductor 1 in Q5
1 z$ B' x# X3 @  Ytrue Order of conductor 1 in Q5
* J$ A  U5 h* A; l- T[
$ i6 Z( b8 G' E. s5 i0 z9 h6 T' ?    <w - 5*Q5.1, 1>,
+ k. ?' b/ v. Q    <w + 5*Q5.1, 1>
]
8
Q5.1 + 1
$.2 + 1
, q6 m, H+ [' Y$ @, V8

>> Name(M, 50);
       ^
Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]

9 Z2 u' u8 p; ?/ v0 `; D$ X/ ^1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
$ t5 K7 I+ L* uMapping from: Abelian Group of order 1 to Set of ideals of M
1
1
- a0 e( t8 M# u) f1 ^; j/ UAbelian 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
3 c" i  b3 u1 `5 l2 A  D% g9 BMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
! A0 z5 h, R! g8 b8 given by a rule [no inverse]
! u0 I3 T: U& p8 S: {Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
. B7 ?2 |( }% D8 given by a rule [no inverse]
true [ 5*Q5.1 + 10 ]
. f  n) W8 l: wtrue [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

基本单位计算fundamentalunit ：
5 mod4 =1                                              50 mod 4=2
  L* m( _) i" D1 f" M$ \& c8 W7 a
x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.

% d4 _0 w- M0 m8 ?5 {% ^
最小整解（±2,±1）                              最小整解（±7,±1）
                                                             ±7 MOD2=1
2 y0 ~, E6 `7 M0 d9 R6 S
# X9 p, C1 h# _; U2 {: i: {- D两个基本单位：

lilianjie1

lilianjie 发表于 2012-1-4 18:31
基本单位fundamentalunit ：
6 B% e0 h. n' p* X* T/ A/ _: ~5 mod4 =1                              50 mod 4=2

4 v) m( f- |. h; T. {基本单位fundamentalunit

lilianjie1

- t/ L! q/ l' Y7 T) v% r
' {3 }7 i# D) o/ ~3 {判别式计算Discriminant

5MOD 4=1

* Y7 T2 M; a# d& h（1+1）/2=1          （1-1）/2=0

( k* a# j" _+ Y5 C# i+ j4 a% SD=5
* w2 f8 U3 K$ k( f

50MOD 4=2
D=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44

# z! `  X1 T  b2 P3 E# n分圆多项式总是原多项式因子：
/ D3 H0 V9 Z0 u8 _7 s+ U* pC:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

' [0 X3 {' S& M
分圆域：
分圆域：123
' l& Z6 [0 s% l! r: a' J1 I- C
2 y1 L# T. ?+ `+ jR.<x> = Q[]
F8 = factor(x^8 - 1)
$ h. z3 q6 b1 f1 d# q, k1 A2 pF8
7 V# o7 Q* {3 z) U& j9 h' x
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
8 O! D/ d. h) f1 x
Q<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;
) u* C& _* V/ r' z- w: ~: ]0 x
F := QuadraticField(8);
F;
D:=Factorization(FF) ;D;
( C* X4 X8 t# O$ m3 I4 A1 lQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
( C" I# Z$ l% s. n0 T9 Q$.1^4 + 1
6 \" L' V0 Y1 GQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
4 l- Z; }7 _* T; q    <$.1^4 + 1, 1>
]

R.<x> = QQ[]
* ?5 u; j  E- m, ~# ]2 |  hF6 = factor(x^6 - 1)
F6
5 m  @6 v9 f$ q# E8 E- m% l9 b
& E/ K8 q2 _8 t9 b(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
* ^2 v6 W& {, k, i
/ j9 l& J6 Y* @Q<x> := QuadraticField(6);Q;
# C" F" [5 t4 n5 f+ @( HC:=CyclotomicField(6);C;
9 V6 S2 w+ a' \" j8 m2 ~& p: hFF:=CyclotomicPolynomial(6);FF;
% r( W# t1 Q* m/ J5 M) h+ L% M. `
, Y$ ^9 v. y$ \3 U) AF := QuadraticField(6);
; ~) \* l# b" a$ w1 xF;
+ v  N/ \/ X9 `+ q2 n3 ID:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
. L; R0 v$ T7 S& B% p* ]* aCyclotomic 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 - $.1 + 1, 1>
4 `$ p; h4 J" k0 U5 a. S- F]

R.<x> = QQ[]
F5 = factor(x^10 - 1)
F5
3 \9 c( d" b. h8 h/ w(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
  k4 ?/ S# k' W4 Z0 J1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
! X$ R$ X9 O8 S/ J& ]
Q<x> := QuadraticField(10);Q;
# _) ~3 s  F# EC:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;
! k. F8 Y- w9 a& B
& t* u5 W8 g' C6 Z6 SF := QuadraticField(10);
: y" G6 r+ c2 d' \3 ?9 A* GF;
4 J. p6 {- K) T( HD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
# w! T. z) w* z* b' c  x9 a( YCyclotomic Field of order 10 and degree 4
) E0 m% c5 ^' e1 M! i$.1^4 - $.1^3 + $.1^2 - $.1 + 1
' {0 m5 k+ u  Z& a) n) G+ eQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
5 o3 s8 ~. t- X, w3 R( E# b2 P& K]