实二次域(5/50)例2

lilianjie

. \3 D. u; w- l- L( bQ5:=QuadraticField(5) ;
4 ~  [2 N1 i) F; \; YQ5;
9 c! f$ w! ~, ?- B# Q7 vQ<w> :=PolynomialRing(Q5);Q;
7 b  M: n/ @9 Y! E. z
# R5 N* i0 G( L3 @2 X( rEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
. U* C5 y- Z) G# ]* [' mM;
* c8 t$ d& ^, qNumberField(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;
( Q  @% p2 t7 V( ^7 Z# YIsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);
# \; U  b0 Y, m) I% d9 j9 |Factorization(w^2-3);
Discriminant(Q5) ;
" y: [+ ?) h+ E* \% eFundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
" d. s) t, _+ q: }# ^" T* p6 WName(Q5, 1);
Name(M, 1);
; t3 d- f+ ?( B( [, I; R" SConductor(M);
! Z4 ]0 L% N  L9 Y) m2 bClassGroup(Q5) ;
" g7 r$ X$ \7 j3 l- DClassGroup(M);
7 I+ k8 I* Z1 j7 E/ l' RClassNumber(Q5) ;
ClassNumber(M) ;

# b% `4 }4 p9 r1 j6 F; P. SPicardGroup(M) ;
7 b4 _0 C6 z5 i2 L$ w: u/ v3 LPicardNumber(M) ;


# B- P2 X0 A( m4 G0 `, PQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
/ U/ [2 E, z! v# z, J, Z$ h% u

NormEquation(Q5, 5) ;
8 T- a) E2 E# V% UNormEquation(M, 5) ;

/ A% ?0 a, K$ R4 M# Q! p% E; R
/ k& z* p/ p  j6 QQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
6 k( W! y' y5 x, d/ _& _Maximal Order of Q5
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
7 \# t5 h- [9 y8 h+ ?: I" r% }) h, G. }Order of conductor 625888888 in Q5
; I$ v& y/ r( g7 N4 P0 M% n6 htrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
true Order of conductor 16 in Q5
- Q1 E2 q8 R& s! ^3 E9 U5 Rtrue Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
[
* o( t( p& }7 q% B    <w^2 - 3, 1>
]
' i4 ^/ K) v8 Z6 G5
1/2*(-Q5.1 + 1)
+ e, c5 \; K6 ^/ ]* u-$.2 + 1
5
6 p2 W' u  P! E. I8 ~* N; |: fQ5.1
$.2
8 Y# d( ?4 ~1 t3 `4 V6 V8 m7 e2 S1
' |0 [  y1 A* \2 y) hAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
4 s: i9 J" I; R2 tAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
$ M+ k* O4 [+ n. p7 D1
Abelian Group of order 1
  Y; W( I; p. H5 d# mMapping 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
3 y- j. @8 z" Z6 v5 given by a rule [no inverse]
, m0 @4 b9 k. `* |4 O* g; |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) ]
5 E' ?- L0 L% w5 ^) Utrue [ -2*$.2 + 1 ]
+ X% K7 w; ~6 Q. X' b





. w0 G" S# Y+ q( b% A


/ r- @2 I4 w& d+ R$ l

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

Q5:=QuadraticField(50) ;
Q5;
( q! ]$ @) A6 A9 j3 ]
Q<w> :=PolynomialRing(Q5);Q;
8 _: E5 e8 G; c6 zEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
( c' E/ _5 n8 U0 c: O5 m- Q: xM;
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;
. A& b& F- o) M. b, R1 Z' QIsQuadratic(Q5);
) f* U9 ?7 g5 a, n9 m" ~9 aIsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
1 F, Y# l6 p! L; m: y( b6 mFactorization(w^2-50);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
0 g; t% O9 t1 O9 V# i% ZFundamentalUnit(M);
Conductor(Q5) ;

$ t8 N0 e4 c# ]* {* HName(M, 50);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
  D7 x; s9 W: e: CClassNumber(M) ;
, R, h8 s# ]; k  ]6 o8 T8 kPicardGroup(M) ;
: w" J8 }; o& a3 A: h$ D- ePicardNumber(M) ;
7 Q: p- }' S$ \* J+ P. Y  ]
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
/ \; {2 @  V$ W8 eNormEquation(Q5, 50) ;
NormEquation(M, 50) ;

Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
4 `+ V& f! K+ f0 {5 _Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
0 O- O0 k7 i2 y, d7 G0 vQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
! i! r4 w6 ~* u" i" t2 C& d. H0 VOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
2 h/ B8 F: }# `1 W6 H: D, {true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
) @. n& j7 T2 K[
    <w - 5*Q5.1, 1>,
3 r: d2 u) t+ g, v5 Z: C    <w + 5*Q5.1, 1>
]
: ?. r; v  _9 p& @0 `: X8 Q8
Q5.1 + 1
& t8 E5 Y" U3 y5 c- j$.2 + 1
8
; Y0 S' h- S  M. e
>> Name(M, 50);
, u  S6 G" q: f; ]- ~       ^
! K& P1 r4 g, NRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
, D. l  l- `9 y
1
, z( c9 W3 g  ~- u4 e- XAbelian Group of order 1
, y) Y( O( K$ s  `8 YMapping from: Abelian Group of order 1 to Set of ideals of M
$ j3 F! v# u+ R) e2 jAbelian Group of order 1
4 [" P6 ]/ I6 w. I# TMapping from: Abelian Group of order 1 to Set of ideals of M
1
, w5 V* O! p$ l5 e+ H1 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
; b$ W1 `  m3 j1 |  H+ hAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
/ F4 w. e' {1 A1 L7 {( nAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
1 m$ ]% G# r" n6 k8 given by a rule [no inverse]
5 Q! r2 p2 z2 |' K# P$ Ptrue [ 5*Q5.1 + 10 ]
7 P! R& X0 `. j* N, h: c- |( ?2 rtrue [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

+ `; n! o9 b' F2 o3 D9 F0 R( ]
基本单位计算fundamentalunit ：
" h3 {" |1 }: p2 Z5 mod4 =1                                              50 mod 4=2
% p1 [4 c  B% u& o% L
3 B8 ?# O4 H! ^" b; c  H x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
, b) J' q6 ^4 G3 ?2 e x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.
' j; S1 R2 s0 u3 ~9 s3 h) F

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
  A- K6 U; @: p8 A% s* H基本单位fundamentalunit ：
5 mod4 =1                              50 mod 4=2

% o% C/ }8 m1 q8 g# G5 l基本单位fundamentalunit

lilianjie1

判别式计算Discriminant

4 W& `1 h, j! a$ X& s0 h5MOD 4=1
) P' t8 X( `5 F) h# Z
/ f5 l( k# ?# j/ q3 R（1+1）/2=1          （1-1）/2=0

D=5

: h! @. |; |" ~$ h+ b
50MOD 4=2
D=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44

  M5 f8 m- l; _$ h% J分圆多项式总是原多项式因子：
: n. Z3 l0 b: J! ~: D4 j, }C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

分圆域：
分圆域：123

, ?! q4 _, V; QR.<x> = Q[]
F8 = factor(x^8 - 1)
F8
# _$ ]$ R$ [  Q3 L+ Y7 b4 h4 S" [
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

# B( D% u/ _* }2 @' s2 Z. N1 RQ<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
! [( E/ d  C) b+ qFF:=CyclotomicPolynomial(8);FF;
! ?' m3 h9 V- Y( H
F := QuadraticField(8);
$ s. |* K% L" d* [5 sF;
D:=Factorization(FF) ;D;
& Y8 b& z: w( t/ XQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
$.1^4 + 1
. V8 E8 G& ]; k; C' {1 {: ^Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
5 _6 u& E; v7 N" @7 q% U: s4 s[
: i2 Z4 ?9 T5 t8 N' @  T! ]& ^4 o    <$.1^4 + 1, 1>
]

R.<x> = QQ[]
' L3 g- \$ S: H' \F6 = factor(x^6 - 1)
F6
* X( m0 S4 _& a! J. p6 U( j8 i
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

Q<x> := QuadraticField(6);Q;
& O( k$ L; K) p4 Z% `2 k) FC:=CyclotomicField(6);C;
2 l: s9 k6 `2 V+ y: CFF:=CyclotomicPolynomial(6);FF;
+ l2 M% I$ c- G! Z8 x* S. k3 w  ~
* H/ d! l) @) |' ^5 L! e& k/ \F := QuadraticField(6);
F;
7 c8 P. h9 k9 j3 _8 iD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
; `4 E, Y: R0 E/ N: D' {Cyclotomic Field of order 6 and degree 2
! D8 Z& [/ A7 O8 R9 Y8 f$.1^2 - $.1 + 1
8 V1 L  B- f6 y+ BQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
" {# q4 V1 w  w4 t+ i[
) N5 i8 i, U- V2 s& t    <$.1^2 - $.1 + 1, 1>
]

  |. Z1 v4 h/ _/ _% GR.<x> = QQ[]
9 D9 S* @3 V/ \3 `  KF5 = factor(x^10 - 1)
% O  _9 [9 K% j% y' x* MF5
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

- z% v) x& @0 x8 ^Q<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
* h! k6 v* e" V" o3 M- z" }9 eFF:=CyclotomicPolynomial(10);FF;
. {! h) q2 b* b* ]% E! |9 T9 x- {4 F* P
F := QuadraticField(10);
% o9 V& Q* f2 P1 @# O5 x5 Q1 O' sF;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
$ O; K9 l1 F0 [. A$ e0 eCyclotomic Field of order 10 and degree 4
$.1^4 - $.1^3 + $.1^2 - $.1 + 1
1 ]$ G  f1 \4 Z6 e3 E3 k- BQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
' ?9 P, I( U. x# ?    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
+ C* U! K# a) C]