实二次域(5/50)例2

lilianjie

Q5:=QuadraticField(5) ;
Q5;
Q<w> :=PolynomialRing(Q5);Q;

# H# `6 k8 H" c( [( aEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
NumberField(M);
; Z9 h& C& d* V' _- r1 NS1:=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);
% r2 c, V2 ^/ ?" y. h* w% ?9 }Discriminant(Q5) ;
0 j+ T; e( G4 MFundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
Name(Q5, 1);
+ B# ~  k, c% KName(M, 1);
* ]" N7 k" G5 Z3 I2 b& YConductor(M);
ClassGroup(Q5) ;
4 a. y( S6 D% W0 m, ZClassGroup(M);
* m# a! `8 L( }# r! D  s/ }! qClassNumber(Q5) ;
9 {* G0 L( R5 O4 J+ g" E5 b  F* M3 |ClassNumber(M) ;

PicardGroup(M) ;
' o: Y  ?) ^+ O6 oPicardNumber(M) ;
; ]- j( q  d+ B; A( M5 Y

2 N  r$ O8 }* D( V2 m( X! C7 y' u+ pQuadraticClassGroupTwoPart(Q5);
' p8 e9 {* R: _/ nQuadraticClassGroupTwoPart(M);


2 L# `. ]' U1 f" c/ sNormEquation(Q5, 5) ;
NormEquation(M, 5) ;


5 G. ?# n/ F. Y, d7 Y- B- GQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
! d& W$ r/ x  I4 U% q3 J% nUnivariate Polynomial Ring in w over Q5
$ R3 S$ w" @* C* eEquation Order of conductor 2 in Q5
Maximal Order of Q5
# ^( r$ l% ?# N7 x% tQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
: q/ ~7 e$ J# Q! f0 n+ mOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
9 E( K0 e: N' f7 }( gtrue Maximal Order of Q5
true Order of conductor 16 in Q5
' ~8 A' [) h9 |" [8 Strue Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
" v2 M" Z) F( B2 t, h[
$ r0 b/ n' v" N* \' S$ h2 C7 z    <w^2 - 3, 1>
+ [" L2 S: r7 y]
/ z' P5 o% N. Z9 i# N- }5
; r! J, I0 Q6 e+ Z3 P1/2*(-Q5.1 + 1)
: S# i0 ?' y. {& |# \: S3 c# L-$.2 + 1
7 I( C1 G" `" w% f5 y5
Q5.1
$.2
" g  R! i7 z# t. r1 b2 x& V0 u4 `1
Abelian Group of order 1
" W/ g! G( T6 o* gMapping 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
  n) U% f  l  o- i- b( V9 b' G0 [1
$ k/ a7 t- F- T% Y$ cAbelian Group of order 1
+ s0 h& t9 X6 n  {0 [* bMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
Abelian Group of order 1
7 Q/ a: Q* G  q$ b) @4 e6 nMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
' |- F. ~5 s& f/ G5 given by a rule [no inverse]
Abelian Group of order 1
3 V! X4 Y. i( w( I) B6 y& t' dMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
! ?, D7 J! X7 p3 e& s5 given by a rule [no inverse]
true [ 1/2*(Q5.1 + 5) ]
1 W. u' f5 f! j# ntrue [ -2*$.2 + 1 ]



" V# q4 s6 V+ {; s9 S7 y

/ d% u1 Q- w& t

# Y0 V( L7 f9 w) K4 U

$ S5 K) J$ ]- v6 U! y# m
& W3 B& B# x2 E( F7 x1 n' O* e: q
==============

Q5:=QuadraticField(50) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
. _/ z, B% T0 c7 `- t8 m, _# ^EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
. _. p! N$ {( Z: n* ]M;
1 ~2 N% l- d- P' MNumberField(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;
, E0 `. g) {% K. _IsQuadratic(Q5);
3 j" Y( z- @. ]; p% SIsQuadratic(S1);
( @9 ~" s/ B7 J' u4 K8 [8 AIsQuadratic(S4);
IsQuadratic(S25);
) c& J+ N0 |' I7 m4 n; u( |IsQuadratic(S625888888);
Factorization(w^2-50);  
% }  G3 u% Q9 rDiscriminant(Q5) ;
3 A# t, e0 L- }* p& RFundamentalUnit(Q5) ;
FundamentalUnit(M);
' s" v* F$ N$ M' U5 f) @* dConductor(Q5) ;

Name(M, 50);
Conductor(M);
' T, [/ A9 [+ U6 fClassGroup(Q5) ;
! |5 N) ~6 H& S% w+ @+ l1 Y6 ?7 QClassGroup(M);
% }* }* o+ `; U4 [' n2 {# bClassNumber(Q5) ;
( H3 J. \- z* M3 J' m, KClassNumber(M) ;
4 k% e7 }$ x# }. F' _PicardGroup(M) ;
PicardNumber(M) ;
7 g9 n1 }, d) G2 Q
QuadraticClassGroupTwoPart(Q5);
; R4 ?3 Y  |6 l  }3 }' kQuadraticClassGroupTwoPart(M);
NormEquation(Q5, 50) ;
- M  ~* p, W0 X' c! l9 T% X% iNormEquation(M, 50) ;

Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
* P! i& i6 \) g- d# ^3 [8 y  |Equation Order of conductor 1 in Q5
2 ~9 u+ W& n' }# r8 u" o) e' ^Maximal Equation Order of Q5
5 C+ p( ]: L' t( @3 z7 aQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
" u$ _. P7 L! v) rOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
, ]0 R; l; O4 o. Y4 `! m% vtrue Maximal Equation Order of Q5
! p9 S3 K( d( v/ U. u+ rtrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
* L( {1 o) }! F$ ~# D* Ktrue Order of conductor 1 in Q5
[
    <w - 5*Q5.1, 1>,
% A* f2 m2 C' n- n. \5 s( I    <w + 5*Q5.1, 1>
6 p, q% F2 @; O+ J" q( V]
# d+ b  |! [( c1 P" e' {1 @8
Q5.1 + 1
$.2 + 1
- v# s3 l. @: B4 j( ]: I  T8

- ]. A! }( a, j+ C3 W>> Name(M, 50);
       ^
Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
3 i; j8 ^- I2 h" n+ n- i& r
0 A' y1 f! ]  s7 f  H# |1
: w; ~6 m, a- `" f9 C- e9 S$ nAbelian Group of order 1
( E* K+ q" |& x  ZMapping from: Abelian Group of order 1 to Set of ideals of M
$ o8 b" b4 Q6 u( yAbelian Group of order 1
2 n6 V) b! C2 [0 D0 e& [Mapping from: Abelian Group of order 1 to Set of ideals of M
; j. }7 K2 a) u' H. Z+ |5 L" E+ m1
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]
1
Abelian Group of order 1
. j" q+ [$ g* C, Z/ N0 UMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
6 f' p" W; E, ?6 F5 n6 C& y8 given by a rule [no inverse]
6 `; J, W" R( G) e6 l2 ZAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
% }( `! f9 r% Q2 T) rtrue [ 5*Q5.1 + 10 ]
7 E. v& A+ }+ g) Utrue [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

基本单位计算fundamentalunit ：
) G* ?: x7 M% t" _( |5 mod4 =1                                              50 mod 4=2
: C; z( c9 c8 T7 u+ a) ~- A
0 }% D7 b4 C) X  {* n( p) ^ x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.


最小整解（±2,±1）                              最小整解（±7,±1）
  D6 h4 Z7 q/ i) S4 g" Q; P' I                                                             ±7 MOD2=1

两个基本单位：

lilianjie1

lilianjie 发表于 2012-1-4 18:31
基本单位fundamentalunit ：
5 mod4 =1                              50 mod 4=2

* l9 s) d; b7 f) `2 }% Q基本单位fundamentalunit

lilianjie1

4 I% v+ ]6 [* \
判别式计算Discriminant

5MOD 4=1
  E' Y2 s1 K$ r9 O
$ x7 N% n! X/ G3 s4 G+ v（1+1）/2=1          （1-1）/2=0

# E! x$ y5 W+ H9 Y; J* R! I) ~D=5

: ]4 l( w, O4 G2 D' [5 E1 C2 I
& y5 @- u5 s( ~. g( R* q$ \50MOD 4=2
D=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44
3 W; ^, {8 o4 A& w' _& T& W3 R7 [
分圆多项式总是原多项式因子：
" ^8 M1 s& c0 f0 f* _* lC:=CyclotomicField(5);C;
& L" s; t! i2 ^. e2 I+ l0 sCyclotomicPolynomial(5);

3 e+ E/ M- X! Q
分圆域：
分圆域：123

( @5 }6 Y3 @0 l' t3 DR.<x> = Q[]
& f6 S% m% X" cF8 = factor(x^8 - 1)
F8
8 @: |) J% H1 Q% h# r/ o' Z: x
4 ?# g: q# v3 `(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
, @9 t% {- W2 [$ a
Q<x> := QuadraticField(8);Q;
! J9 _% a, Y; J. AC:=CyclotomicField(8);C;
" H6 j( k- Y  C4 v+ b, kFF:=CyclotomicPolynomial(8);FF;
/ G0 d. p; V, q
6 B* ^5 q# k" U4 Z" ?4 K2 @F := QuadraticField(8);
  `( P: V* ]. y% ~F;
$ c2 w2 U2 f* D  s1 TD:=Factorization(FF) ;D;
- a% ?+ c, Y! m, cQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
$.1^4 + 1
- z1 n7 ~& _8 {. V; }- k1 |Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
    <$.1^4 + 1, 1>
) U) ]( D  v1 g9 o1 k5 J* Q]
1 l# f7 N9 Y; G1 q9 G
R.<x> = QQ[]
% Z4 H$ K6 I3 b2 _* A2 ]% zF6 = factor(x^6 - 1)
F6

! S$ j  ^- J: \7 S(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

+ E6 Y+ t; w7 oQ<x> := QuadraticField(6);Q;
# M0 [4 M3 U4 G/ d0 VC:=CyclotomicField(6);C;
7 ~) }; e' E$ J( ]FF:=CyclotomicPolynomial(6);FF;
' m' l% R& Q+ \- h& W9 s$ \7 x
F := QuadraticField(6);
F;
5 y/ M9 P& v6 RD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
# p$ _- M0 j8 @$ f' V; S1 NCyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
, ?) e% {; d+ B7 g2 V; p7 f2 r' D) tQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
: w$ r8 b0 D3 b* O9 A( |( R' B[
    <$.1^2 - $.1 + 1, 1>
2 Z" H0 C% L7 C. Z$ e$ D7 _]
7 c6 s4 C/ d# i1 Y% Z
  p. Z6 l. A7 T0 wR.<x> = QQ[]
F5 = factor(x^10 - 1)
1 {" E  I( n+ o4 y  XF5
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
  Q- H4 g5 E0 [! F! p/ d9 X( Z1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
1 t  `/ l2 x: s5 q# C
: @9 |1 q4 K7 L3 F7 fQ<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;
( p) o9 e. i# ~) l) r
F := QuadraticField(10);
F;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
" O: m. w  y' V, X% t% E! x" gCyclotomic Field of order 10 and degree 4
! p- m* A5 W* I9 K$.1^4 - $.1^3 + $.1^2 - $.1 + 1
. Q* I+ g( u- |4 o' v2 P" n/ l* J5 g6 TQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
3 E. \) @! T' U7 ~# T[
) y) t. D8 T: u8 D* s8 r: @    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]