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标题: 实二次域(5/50)例2 [打印本页]
作者: lilianjie    时间: 2012-1-4 14:05
标题: 实二次域(5/50)例2
本帖最后由 lilianjie 于 2012-1-4 17:59 编辑
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4 ~8 t/ V- J& ^+ A! UQ5:=QuadraticField(5) ;
+ a: J0 G3 X# W1 tQ5;% u! [, b2 y; x
Q<w> :=PolynomialRing(Q5);Q;7 u2 R+ ?5 v1 r) a# C' K2 K) d1 G

: k  x; M; P3 n4 k5 L* b& b# q& @EquationOrder(Q5);6 p6 Q. ~: `, ~/ N( q; {5 T
M:=MaximalOrder(Q5) ;
9 ^5 H) H+ M. h* ]' p" p# \M;5 L; n9 b5 B5 e* Y
NumberField(M);
4 F% B; o' z& Z! q. p3 s- HS1:=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;: x$ s9 _8 Z3 P" v3 B( Q
IsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);8 |6 z$ s6 W4 W2 r0 v. q0 i
Factorization(w^2-3);: T* y1 |6 k0 |4 C1 m
Discriminant(Q5) ;
! d% p3 H* S. wFundamentalUnit(Q5) ;! Z6 B' Z. ?# v) ]7 h
FundamentalUnit(M);
1 ?1 e+ v7 |" y3 g3 a  P' n# |Conductor(Q5) ;( l, D3 |: N; }1 D% E
Name(Q5, 1);
3 E( b% d0 a2 C* X2 D+ RName(M, 1);: F& u1 }+ ^$ Z! q- H' v  s; w
Conductor(M);) M$ c% d7 u/ e  p. j0 g2 \
ClassGroup(Q5) ;3 o6 b' R8 H8 a; W: c4 i
ClassGroup(M);  Q8 V. S5 q6 k9 v5 n
ClassNumber(Q5) ;3 X' M- t' u2 i! Q9 s
ClassNumber(M) ;/ H5 q* Z: K9 Y2 o- C

2 q7 t# N, M  b: F$ X+ uPicardGroup(M) ;! ^* ~" W. W3 p: S
PicardNumber(M) ;
5 y1 s# x4 A/ e3 x$ K4 Q
- c+ m* M9 a4 C
( S, T  N, I, n, Y' T. FQuadraticClassGroupTwoPart(Q5);
. V& N  ^1 x7 K( Z, D% P/ n- hQuadraticClassGroupTwoPart(M);" f* ]  }: J9 |- W! |; e: Y) R
: i2 ~8 Q0 {- S2 q' W/ J; u

, @2 c9 u# o& R; K  S/ l% w( w( ^7 V' sNormEquation(Q5, 5) ;
; P6 I7 \3 j7 {4 s( rNormEquation(M, 5) ;; P" c8 E/ V- C. X! m. O4 I
' a. h- p6 `$ `7 n& [! |
% L8 y9 T* G3 x! i0 d7 }
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
/ S/ ^9 z; o/ D+ ZUnivariate Polynomial Ring in w over Q5
. p$ ~! n0 A% [, M+ u& vEquation Order of conductor 2 in Q5
7 {& q$ T3 n5 _' J8 EMaximal Order of Q5
0 s7 F3 K$ @* g$ t& p/ W& K" k. o8 Z% ZQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field8 w0 M0 w/ v/ [$ c* \( q) e, A
Order of conductor 625888888 in Q5* R7 ~  @, i6 t) n& }1 s
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field1 s+ |$ n" h& s  w! w" C6 q
true Maximal Order of Q5# Z2 y' e8 z/ \5 T$ L
true Order of conductor 16 in Q5
* t0 a( m4 ?: B/ M& k" ]+ O$ q6 Strue Order of conductor 625 in Q5. z  M0 r  T. `3 H6 S9 ^
true Order of conductor 391736900121876544 in Q58 k. U! Z6 A( B; \
[: p7 ?8 e4 U- h" ^: }  n6 Y4 B
    <w^2 - 3, 1>
& {' ~/ [$ M& E], {. i  F- ?5 [) F
5
' {; p# s! k7 R2 y2 s, n1/2*(-Q5.1 + 1)
+ r0 X5 b+ [3 j8 [0 h8 O-$.2 + 1& v4 Z% }* S# |
5
+ q8 c8 Y1 Z9 r* ]Q5.1. Z# ?4 j) S" A7 n
$.2
! o1 M3 v1 w( X, F& B, V% N4 O) b( I* _1
9 p% s, _3 H8 CAbelian Group of order 1
4 Z% @1 w6 \/ f& J- a6 B) W% }Mapping from: Abelian Group of order 1 to Set of ideals of M
# n* p+ n9 U* R7 X) H. G8 UAbelian Group of order 1
  l- ^; e* E( z, `# S2 ^Mapping from: Abelian Group of order 1 to Set of ideals of M* u; P8 X8 W8 A; G" y
1
5 F3 E. \' V5 q( S- J1
% {2 k2 X3 a/ y) GAbelian Group of order 13 m7 N4 [# \. h* T4 b& s" N
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
/ I' \& i9 Q$ W$ e6 minverse]
" s! a2 @" c1 v9 e1
) y9 h9 ~6 r0 k  d7 F+ }0 bAbelian Group of order 15 l$ c8 R4 x) U. a* v
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
, l  v5 K  A, ?. \5 given by a rule [no inverse]
! E5 x+ F2 R2 a" W1 A. ^" QAbelian Group of order 1
' {- j' A3 T3 u6 p' X; [5 ^Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
* o5 @& l0 i0 ?2 ^5 given by a rule [no inverse]
7 C4 o& p& M2 k. Q$ n0 l! btrue [ 1/2*(Q5.1 + 5) ]. Z$ @7 B5 Z4 H5 c! N$ d5 Q
true [ -2*$.2 + 1 ]6 I( N" L1 e( e1 E9 X+ X0 W

  `! t0 J1 m: u1 T! b: j
1 ^- z7 \3 g! a, X! u: C8 A$ q7 L; S2 d# o

: E7 g2 f- p$ p& _' D1 t  y6 V" @' |0 \5 B# H
* z1 p* U. ?4 `' A5 T
6 l% \, {7 G5 P1 ?9 W
  p! L2 m4 h" U& z
6 o4 K2 U  h; e, l& H- u- Q6 m4 A
% Y- {. ]: J, z" x
6 U3 m3 y. M$ A! g- U+ ]
==============
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' p. u) R7 F" r: k$ f4 f6 K5 x' H0 eQ5:=QuadraticField(50) ;2 |# q0 s/ n1 `4 A, E
Q5;& R3 `. J" O. H# N+ a

: e# ~' c% h: HQ<w> :=PolynomialRing(Q5);Q;, H" H1 f; O1 v+ t8 }
EquationOrder(Q5);% }8 g2 O% B  o- v  O1 _2 d: g, l
M:=MaximalOrder(Q5) ;
5 f/ P3 N& N7 w. M" p9 DM;
$ |0 P6 V1 ^! V( vNumberField(M);+ r. C5 r4 f8 E% r6 N2 n7 k
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;
7 b$ w4 U) w' F- [8 \# |IsQuadratic(Q5);9 x5 z! J: a; g+ w
IsQuadratic(S1);& s  d- b' n' z$ T7 H' u- O0 Y/ \
IsQuadratic(S4);+ P) h# k3 l3 r( r0 h; X+ i2 V4 f
IsQuadratic(S25);2 N. f3 T2 B' Y( z% I* a6 o
IsQuadratic(S625888888);
, Q7 b1 b/ M( I  u8 R3 hFactorization(w^2-50);  
. n) [+ ^" t! m5 n2 ]# m# F- bDiscriminant(Q5) ;7 W, ^3 u# a% R: P& ~( W
FundamentalUnit(Q5) ;
( k: b: v& h6 v! W) {2 m2 R# AFundamentalUnit(M);: _" P9 A( h, O$ g" o
Conductor(Q5) ;/ j' ^" y4 W; F: |( K0 H$ j3 _

. Z5 G  Y! G% Q2 _* [' y' [Name(M, 50);
5 d* v8 }4 U7 z  P. p8 f4 LConductor(M);
- m2 j* `% I( }2 z8 k; RClassGroup(Q5) ; / r' w" W) H7 o+ I
ClassGroup(M);/ b2 R7 Z' {& z; s: o
ClassNumber(Q5) ;5 {- \% V' x; r4 e+ s3 k
ClassNumber(M) ;& [( X$ D+ ^) Q1 |) U
PicardGroup(M) ;
( ^3 N# q# V* O6 p1 L& i' s: f! M$ f' RPicardNumber(M) ;
, y1 Y0 b5 T3 {; A
# P4 c2 z& ^# N0 j$ y' f% D9 UQuadraticClassGroupTwoPart(Q5);! R6 O- o9 I) \  o. `
QuadraticClassGroupTwoPart(M);, k& ?3 y5 h+ x( i6 G  M9 D
NormEquation(Q5, 50) ;2 z& q: H+ R% D9 G, }0 q& Q
NormEquation(M, 50) ;  d8 O$ N- R! O0 a

0 c0 C& M- u6 h" O, ]$ `8 ?Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
; q: K  Y2 _9 h6 P9 AUnivariate Polynomial Ring in w over Q5" d* m# x8 K% o4 N% }
Equation Order of conductor 1 in Q5- B3 t% t* K  a0 Y
Maximal Equation Order of Q58 A4 e& H9 z+ j  u& ]8 e
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
% ]! U1 W2 B3 n7 j4 r2 x* cOrder of conductor 625888888 in Q5
) }) I9 p8 j7 v0 A  n& Ftrue Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field( s8 e! t; I4 m* K
true Maximal Equation Order of Q5& p4 k  H1 m0 E, b2 _' P
true Order of conductor 1 in Q5  Q8 ~7 Q) S6 s/ |* z" n
true Order of conductor 1 in Q54 ^: {6 P0 H- `7 T; W5 n( O
true Order of conductor 1 in Q59 ?) s8 B" T! s! o# f4 a7 W
[
, ^: [" Z& V" K$ k9 u9 J2 H    <w - 5*Q5.1, 1>,+ a0 D0 n( k. j4 z' f
    <w + 5*Q5.1, 1>* T6 W, \* E8 h) w  \& o- P
]6 r6 [  k1 s) ~" m. K
8! c( p* y5 M6 D2 n8 _" i, U
Q5.1 + 1
! G8 Z& A0 _& m% v" \% w$.2 + 1
( e8 E& y5 M' |- f  e) m" x6 X8
0 [" ^, p- a; i3 Z) }( A; C8 Q# V$ [
>> Name(M, 50);* J2 a2 V% m% }5 Z/ `  a
       ^; o; N# Q/ G! f. s
Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]' v# I; P( f7 G; T
9 F, n9 J6 t& b2 w+ a$ d: F
1
) N3 `" p4 G: b6 P2 w  NAbelian Group of order 10 D' t8 H( F3 i& v
Mapping from: Abelian Group of order 1 to Set of ideals of M- c7 s$ e2 Q& P: q" C3 O+ s+ F
Abelian Group of order 1& F" G; ]$ q' i, p: T) T
Mapping from: Abelian Group of order 1 to Set of ideals of M
0 H( M" R8 C2 J( v9 T& _8 S- I$ q9 p* p1) l$ f$ T1 @% n5 p1 O
13 c4 h+ s* a3 y% i2 C% |
Abelian Group of order 16 e+ w# v; }- X+ [5 n2 z' d
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
0 m" ?/ s& ~$ [0 uinverse]
; r- n$ q3 a. P! H6 l, O7 a5 Y1
/ Y8 r/ B: ?* x: Y, e, j8 QAbelian Group of order 1! I  u' m6 y5 w7 S; ~1 p/ M
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant2 @+ b1 H. u( v9 ^  A; C! A7 k+ `$ L; z
8 given by a rule [no inverse]3 v* M4 i3 w, c( l
Abelian Group of order 1, A1 [+ i; ]% B) C% e8 C6 ~
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
* ?2 E/ e' x. x5 W8 given by a rule [no inverse]
* m4 w% P$ w7 w+ p& m, {true [ 5*Q5.1 + 10 ]
4 t7 q. U  h9 y2 jtrue [ -5*$.2 ]
作者: lilianjie    时间: 2012-1-4 18:00
二次域上的分歧理论

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作者: lilianjie    时间: 2012-1-4 18:31
本帖最后由 lilianjie 于 2012-1-5 12:42 编辑 & O) ]/ y" T8 x

0 R; r& G9 A5 `+ H; U基本单位计算fundamentalunit ：
/ ~" ^( h: a6 S& q5 mod4 =1                                              50 mod 4=2
  J! F0 T* O$ B! H7 ?8 p
$ s: O- R" \, z: L1 E3 y* O% G  p x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.' u( m: B0 F& C2 |5 u
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.
4 M: `6 t+ J! Y% `; W$ e- i " v) a/ B2 g: ^" P
3 ]* Y2 Y* j0 _
最小整解（±2,±1）                              最小整解（±7,±1）
+ T0 C/ Q( k% X( Z4 j                                                             ±7 MOD2=1
9 M6 g. C7 K9 n$ L9 ]! r$ R: W9 M' o/ H
两个基本单位：

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作者: lilianjie1    时间: 2012-1-4 18:53
lilianjie 发表于 2012-1-4 18:31 9 b" |1 |) g: ]+ G) B. G/ H
基本单位fundamentalunit ：
3 I$ I8 j0 L) z3 D5 mod4 =1                              50 mod 4=2
' m' l; s, ^0 `) l$ G/ A2 S
基本单位fundamentalunit

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作者: lilianjie1    时间: 2012-1-4 19:07
本帖最后由 lilianjie1 于 2012-1-4 19:16 编辑
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- Q# O2 _2 F4 D. A% n( m% {判别式计算Discriminant, u+ K$ f6 `1 t: ^, S

3 Y0 R  w8 k+ N0 e% N# B( v5MOD 4=1
6 x& _3 _; A5 f
0 l6 }( B1 l! E2 x& o; n（1+1）/2=1          （1-1）/2=0
* v" G. Q+ _' _! j
" I) {, y) M  R( [0 p, C) R/ jD=5& [8 K3 n, D9 m! q) x5 E

, ]/ e7 D$ d6 v" w% n
4 A2 B; X! s! j8 W$ t50MOD 4=2
: w5 n8 h4 }! e  iD=2*4=8

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作者: 孤寂冷逍遥    时间: 2012-1-5 08:37
作者: lilianjie    时间: 2012-1-10 17:49
lilianjie 发表于 2012-1-9 20:44
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分圆多项式总是原多项式因子：
4 r6 j9 d5 R0 M% c1 ]C:=CyclotomicField(5);C;: |) Z$ v0 P4 M' D
CyclotomicPolynomial(5);

. v0 R% f4 ^6 |9 y5 `% f9 m1 k( v
分圆域：2 w# b1 D4 H2 }0 X* P5 H/ o4 R
分圆域：123" ~, [3 J. w6 k, T3 i9 u5 ]

2 e" L* b/ l8 G) b5 ER.<x> = Q[]
) G/ _! Q5 G/ dF8 = factor(x^8 - 1)7 j& r; Q% F9 L6 n) R0 C
F8
) S  O& d0 f* h6 g, S6 J6 X& U
. l" k( u7 ?6 o' F2 g(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
9 U# Y% O9 V! ]1 R7 m' y
$ V: c8 ^& ]& u; F% MQ<x> := QuadraticField(8);Q;8 W) ]7 f% T) t0 V& I, b
C:=CyclotomicField(8);C;, n( T9 l5 K& \3 L
FF:=CyclotomicPolynomial(8);FF;
0 \( e( k- r& z- V: d6 ]. u% n" H, I, z1 ]& b) @1 [! S
F := QuadraticField(8);
. u+ L9 X2 n( a8 y$ X9 uF;/ i- N/ }$ k+ Z! M2 K: k
D:=Factorization(FF) ;D;4 g9 J1 F# k" {  D# }
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field! |: t9 i% O1 w0 F
Cyclotomic Field of order 8 and degree 45 R, I" R& Q1 f- Q* o
$.1^4 + 15 Y5 e1 X, H9 s( i" ?+ P; q
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field8 w3 n5 Y7 U. }- I4 b) W
[
3 r+ p3 X& h2 A: M) X# j    <$.1^4 + 1, 1>
5 n+ r& ~3 u( Z5 Z2 U. O5 []
$ d: ^3 |8 o& \/ D0 ]  d6 {
. t( s+ D! R. FR.<x> = QQ[]
# V0 v0 P# g' x  Q2 nF6 = factor(x^6 - 1)4 G+ A3 [* o& u8 T9 d2 G. z& x, U
F6; @% T7 |( L/ n% U

$ Y$ J6 J4 Z# |  M* 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) $ A! J; D; h4 W* _

) R+ f. h% L0 cQ<x> := QuadraticField(6);Q;
5 J! o- C. v0 ^& W. i+ PC:=CyclotomicField(6);C;0 X2 S1 w# ~$ \& M$ Y
FF:=CyclotomicPolynomial(6);FF;
8 M1 @( g" G* T! }5 y6 k6 u1 D2 ~& w' y# H; [) l& i
F := QuadraticField(6);
( T2 K! v8 U, x( Y4 n) F6 JF;% T" C, r# E4 ]+ J
D:=Factorization(FF) ;D;2 P2 q2 J, D; f$ _2 R
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
# ^% [2 @. Z( xCyclotomic Field of order 6 and degree 2" F* `2 x) b: o0 U* M
$.1^2 - $.1 + 1
( g! Z6 O! D! Q9 NQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field- H( v5 s* ^. g4 C. e
[4 C8 M- u7 |' B& [" q& v0 N
    <$.1^2 - $.1 + 1, 1>7 s, c! }5 k2 @/ |# x
]" \6 ^# N, \# B0 G- `+ d: |: j

! C+ X6 L8 ~, V$ A7 i. I; vR.<x> = QQ[]* K' J8 Z& P- C  y
F5 = factor(x^10 - 1)
4 Z6 X( U2 @, p3 WF5
+ y% H' I6 Q9 M# O9 c(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
$ H1 Y' Z. ~" S7 g1 V1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
- h2 ?1 D1 S8 m
9 r$ F: z4 q2 U; p: HQ<x> := QuadraticField(10);Q;" _1 G) `! P0 O
C:=CyclotomicField(10);C;( P( t: j% m9 V
FF:=CyclotomicPolynomial(10);FF;
; n, b3 E/ }; ^6 i! L
% A& K, u4 ^* L+ e4 d7 G  YF := QuadraticField(10);
4 G3 [1 A$ N0 eF;
$ J# P6 ~6 P9 M3 FD:=Factorization(FF) ;D;
/ {2 I: ~- P" i2 A/ i: |# lQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
5 j3 K  S! }* o& Y' X) `9 H% lCyclotomic Field of order 10 and degree 41 o) T# p  i/ r8 y$ y1 L  G
$.1^4 - $.1^3 + $.1^2 - $.1 + 1
; @/ V6 g0 w/ ]Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
! u8 J. D+ O# I( x$ l8 V[8 ~8 v: P* @, }2 E* l
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
* ]7 S" J/ n* f]



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