实二次域(5/50)例2

lilianjie

1 V. M6 w9 E5 t
Q5:=QuadraticField(5) ;
Q5;
Q<w> :=PolynomialRing(Q5);Q;
$ X' b/ r# ^, O+ T' w9 O$ i  }
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
5 ?; [; [8 p% a- L* Q' ]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;
5 y; R1 h! Z1 ?IsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);
Factorization(w^2-3);
: k5 T( s8 |" i+ LDiscriminant(Q5) ;
6 d, |: s" L: s" i) mFundamentalUnit(Q5) ;
FundamentalUnit(M);
! L$ \! Q1 `, ?6 o0 C5 l( `8 b3 r- lConductor(Q5) ;
Name(Q5, 1);
Name(M, 1);
Conductor(M);
8 I8 L; H/ F" L! c( @2 y% p! W# |1 sClassGroup(Q5) ;
ClassGroup(M);
, l" C; B* i2 K- F* }# UClassNumber(Q5) ;
ClassNumber(M) ;
: l. v$ I. w# J1 A( _8 B. L$ U6 h
* l; |7 F6 h, k! i$ ?) n1 zPicardGroup(M) ;
8 C+ [# `6 y0 k: _7 y# FPicardNumber(M) ;


) H$ b+ d9 c4 G6 R( VQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
# W  ]2 f9 m# A+ ]
% n- f' B8 O0 l( W+ {, H1 \
2 r) F0 _/ i1 INormEquation(Q5, 5) ;
9 @% E  B& `2 q% b* xNormEquation(M, 5) ;


/ `/ g( `" Y: z6 F8 u: ^Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
" c( x# k/ e- A' _5 pEquation Order of conductor 2 in Q5
Maximal Order of Q5
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
* {% K( s% Q; Y; E+ C) ^+ DOrder of conductor 625888888 in Q5
" T) g: z* U- U: r# i1 `$ Ptrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
# T& u, T; B( g9 g# Itrue Order of conductor 16 in Q5
* D3 a) k0 \6 p: Ntrue Order of conductor 625 in Q5
' k  Y$ w7 ?& p  S3 f8 utrue Order of conductor 391736900121876544 in Q5
[
    <w^2 - 3, 1>
1 o5 G7 q4 ^# M- }; {]
5
$ J( A6 G* E7 A6 \" u1/2*(-Q5.1 + 1)
: d- m* P6 D: B6 ]" Y6 J, ^- d, w( ^-$.2 + 1
' k0 Y2 x( I; d  W+ A4 ~5
3 I/ m. D; f7 fQ5.1
! M2 ~3 P) Z$ O* [- B$.2
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
  u8 T5 s% C# v+ \1
) G$ O, S, ~8 p0 r! FAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
/ W0 Q: ~4 A7 q( G% E8 d: cinverse]
( s3 M$ I" h" |, R+ `1
  ]0 Q4 {6 p+ r' n4 b8 Y7 jAbelian Group of order 1
5 V9 y" [+ J& `7 j) M1 MMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
4 K& Q5 y! d$ M( g) Z1 R5 given by a rule [no inverse]
Abelian Group of order 1
* ]7 v( I- Z7 `Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
1 j- l  J/ I$ K5 given by a rule [no inverse]
true [ 1/2*(Q5.1 + 5) ]
true [ -2*$.2 + 1 ]
1 X* s- o! N2 u7 k7 f7 K" X1 W  V

# H" a1 U) \6 z  T( L
1 F- ?5 a2 j: u+ [
, \# @/ r5 L2 ]& Q: d
4 f0 D2 K& _  R! r3 M0 J4 Z


, {. T. C5 j! e$ g) }

+ @& Z' Q# P, P( C' ?
==============
1 @2 r4 W, X: `7 E. `
Q5:=QuadraticField(50) ;
Q5;
; r" T8 V) n& _/ _2 ]1 f/ s* n
6 W/ r, \" |# b, bQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
! M5 |) D% d  P5 ^4 n% RM:=MaximalOrder(Q5) ;
5 U: s5 [( x2 {$ w- z: BM;
% c6 _0 V6 w' S1 z3 TNumberField(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);
  Q& o' \; p; r; t2 ZIsQuadratic(S1);
1 t# ~( `' ?) h+ i# ]IsQuadratic(S4);
IsQuadratic(S25);
) H: |0 D$ d, {: O# U1 V5 xIsQuadratic(S625888888);
6 M  g8 e, L9 k3 VFactorization(w^2-50);  
9 E/ X" U% ^" f) J' UDiscriminant(Q5) ;
* q+ b$ o: T% J( \& _FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

" @. g5 A, L& [& F; A) LName(M, 50);
' L/ B: n3 ]4 {. C% sConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
4 U- r& B) Q5 y/ G% N& c8 E& t  mClassNumber(M) ;
7 c. V& {  _5 @* G2 APicardGroup(M) ;
PicardNumber(M) ;
0 u: x2 w. g0 f
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, 50) ;
NormEquation(M, 50) ;

6 D- o% X; H, yQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
* k- _% m% r4 oUnivariate Polynomial Ring in w over Q5
  u! _5 P0 T; }: t7 REquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
2 |2 a+ m) v" n( f6 CQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
. K. a7 u/ i7 XOrder of conductor 625888888 in Q5
1 O# z3 S# ~& t# etrue Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
# t# S4 [$ B8 m/ X- X5 m7 btrue Maximal Equation Order of Q5
( ~5 d& f# B. W- v: Btrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
& S6 O, f, t4 Z; n# Q/ J8 ytrue Order of conductor 1 in Q5
[
  Q* w- N( r2 X0 [" ~3 y" E- s+ d7 D    <w - 5*Q5.1, 1>,
3 [6 [8 s$ m8 Q: O0 f    <w + 5*Q5.1, 1>
]
8
# K5 n" O$ ]9 D' J! @; ~4 mQ5.1 + 1
$.2 + 1
  Z' w0 W! I$ {4 |& u8
. Z7 s  \# j8 C
5 v6 f( s1 K; H( U7 C( A>> Name(M, 50);
       ^
Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
& [8 v0 u% ~4 d* Y/ L
1
& j0 m. n- k8 y9 c* M7 b& S7 FAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
+ q# M5 G8 g8 [# d) J8 ]1
/ W# O4 T3 @7 S* h1
; Y0 Y8 r9 J+ `: g& ^; g1 x- _Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
. F: f9 h" Y7 D' I7 C) _$ ]* T- uinverse]
1
Abelian Group of order 1
/ l  y0 Q0 K8 @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
4 t' v5 w$ t+ }8 F8 given by a rule [no inverse]
) ~  w7 u, _; V) Utrue [ 5*Q5.1 + 10 ]
true [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

) D, Q  G* e; p1 w) I0 P7 x. B% Y基本单位计算fundamentalunit ：
. N- K  _3 ?9 Z0 u+ l5 _5 mod4 =1                                              50 mod 4=2
, J( i" o8 w1 L$ P# W) H8 B( b4 h5 O
% |3 m% a: F$ F, V7 \& c x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
1 }( u# }/ G- Z x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.
6 ^# Z& R, D8 \5 K  T
4 s2 E* b5 g" p
最小整解（±2,±1）                              最小整解（±7,±1）
& f1 t  A- p: w) Z+ h, Y& x                                                             ±7 MOD2=1
3 E% Y; X' W+ U, e$ P- _( v
两个基本单位：

lilianjie1

lilianjie 发表于 2012-1-4 18:31
基本单位fundamentalunit ：
+ V- n' [. w( n- o. g  F6 m5 mod4 =1                              50 mod 4=2

基本单位fundamentalunit

lilianjie1

& ^( R- H4 y) Z8 m0 W& y" E" p8 q
判别式计算Discriminant
% ]8 E9 v- y0 ]; S
3 s- j8 q& C& c# Z5MOD 4=1

2 G* _) Q2 W; W! r& Z; {8 p* Z/ K（1+1）/2=1          （1-1）/2=0

D=5
  r7 h/ G/ ]: X; z0 n& k
4 @6 |+ Q% t  T! J
50MOD 4=2
D=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44

# D% i/ w' t3 ^% e. x分圆多项式总是原多项式因子：
C:=CyclotomicField(5);C;
7 _4 X: G0 L7 zCyclotomicPolynomial(5);

  d0 q/ |; U# k6 T1 b分圆域：
分圆域：123
) E. Z& x, ~, H. O9 z  g4 n
2 m2 m; r6 g* v9 c& w+ [R.<x> = Q[]
/ a0 I' s4 @" v4 ?# D; a, qF8 = factor(x^8 - 1)
F8

(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
' e1 e2 F6 M! E! S" x1 c7 ^5 |
Q<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
% c2 Z: s1 c# X+ w7 ?# XFF:=CyclotomicPolynomial(8);FF;

& ?. q: i: L+ Y2 g) oF := QuadraticField(8);
& o2 h  L. J8 P8 c4 N' S" M) K! PF;
6 A( R0 W# c! f# y1 o( m. ^/ s0 mD:=Factorization(FF) ;D;
( v% O- P8 k$ WQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
: i& @  N* ?: C' \( d0 r$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
6 m4 S4 C: w) G1 _2 v! \[
    <$.1^4 + 1, 1>
]
8 V/ O. \: |7 q: m' A
: e' ^' d3 `; Z* T" a6 ?R.<x> = QQ[]
+ a) {% M( o1 Z+ m& h- wF6 = factor(x^6 - 1)
7 P5 |+ A& k) v7 X: ^F6
/ m* `( C# z3 ~6 O% ~2 ^
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

. `1 o! Q! w1 ^" Y. c+ }/ VQ<x> := QuadraticField(6);Q;
5 ]" U& N' C: y7 w# Y: mC:=CyclotomicField(6);C;
9 b& ~4 W/ m1 u* QFF:=CyclotomicPolynomial(6);FF;
% u# L6 e/ r, ~8 B6 s$ y2 f
# l1 ]8 J+ F4 K4 v4 Z) c4 L& }F := QuadraticField(6);
F;
3 g5 b* M6 u9 O: w7 D  `+ pD:=Factorization(FF) ;D;
8 v& s) f) ], C6 SQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
$ s& b7 k1 o+ w& y" JCyclotomic Field of order 6 and degree 2
* J, p7 V% K. p/ ?& A/ h$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
    <$.1^2 - $.1 + 1, 1>
]
5 n/ Q% B3 N, s- c: B  {
R.<x> = QQ[]
F5 = factor(x^10 - 1)
F5
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
0 D+ I' Z, Y7 f/ ^7 v1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

! U( V/ j9 D  W2 O) Z4 eQ<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

F := QuadraticField(10);
3 t+ G+ S' p. K* e1 c& ]% aF;
D:=Factorization(FF) ;D;
+ y- z0 v7 h' T2 O3 G) l# N& xQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
8 n" ]$ k0 W3 N" uCyclotomic Field of order 10 and degree 4
$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
  Q" V3 c; Y2 _' X) U: f) w0 k* H  o[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]