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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 6 ]; |9 s6 |& I

3 P( T6 A8 L/ kQ5:=QuadraticField(5) ;
& D  r. ^' g9 Y. t4 \" W2 FQ5;
, q# i- K' V5 O- \Q<w> :=PolynomialRing(Q5);Q;6 @5 [3 D# W7 A2 L2 `5 ^! |

4 q2 v  ~% F' C* Z0 }- Z# }EquationOrder(Q5);3 w' U. X/ W: V0 }) v: a
M:=MaximalOrder(Q5) ;
7 x* ~  n( V( a; K' C5 ^M;
0 Y) O$ N7 q* R! p8 `* Z% R9 E6 @NumberField(M);
6 @% a' f) L# e  w. o: [' @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;/ i$ u" E# z( h; `/ g: A" I
IsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);
9 M, ?& x% }  D' `Factorization(w^2-3);
2 K% t% w+ k/ g# S& aDiscriminant(Q5) ;) t+ _5 V( m1 `9 o9 f& Q. X
FundamentalUnit(Q5) ;
. U7 J; l* Q6 D% o; M* l: aFundamentalUnit(M);
% k* T0 D4 k5 q9 BConductor(Q5) ;
9 m9 G3 O0 T9 k5 E# rName(Q5, 1);
3 v' u7 n  h% {$ ^1 P" ]Name(M, 1);
) N/ n( Z8 u$ xConductor(M);, b' t; M% C$ `
ClassGroup(Q5) ;
5 v  E8 D* J, \/ Z3 [( d2 {4 |6 Z' f0 PClassGroup(M);
8 o$ V. f/ a; u( }ClassNumber(Q5) ;4 |, I( Z1 \& S# h* ?1 m! U
ClassNumber(M) ;: ?( o9 G- o; s0 y" h5 |
2 R0 {+ \1 p( m1 c* k$ w) g: c
PicardGroup(M) ;
" q: u7 i3 R$ `" R' wPicardNumber(M) ;% L9 U  b  c6 o. f" L
/ N" c& M- c. g; r

. U! H' Y0 A# G' ?  mQuadraticClassGroupTwoPart(Q5);! V& `5 r$ d4 P4 l4 ?" }
QuadraticClassGroupTwoPart(M);2 ?9 `. [- g# ^* S# p

" o( Z1 }0 l5 C
* L2 K5 c: u+ E' r: g+ n: I3 `+ `NormEquation(Q5, 5) ;
) b- I) E4 A" l6 rNormEquation(M, 5) ;
- e1 C9 u& L1 {) c) I) G' I$ ~9 m5 Z. G' Z. D4 [
+ N, U! w: G# x9 w* W0 S3 Y- A& a
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
5 A, D1 p( J. _- {2 S0 d+ ~- F/ g% lUnivariate Polynomial Ring in w over Q5
2 L6 ^* G( b  uEquation Order of conductor 2 in Q5
0 K6 e5 _6 H* N/ @Maximal Order of Q5
# S! c6 o+ ^6 Z/ N' g5 Z! JQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
# L6 ^1 f0 a5 q0 e& \3 FOrder of conductor 625888888 in Q5
& _, Q9 `$ A: m) B* M9 Qtrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
2 Z4 f7 F: w. xtrue Maximal Order of Q5
, P4 k( ?. V5 C, Mtrue Order of conductor 16 in Q5
  r' t& m" A- j% g* v$ Rtrue Order of conductor 625 in Q58 Y& m9 x- {  R1 s
true Order of conductor 391736900121876544 in Q5
( T: n2 q; M$ S% [[6 }* y2 @, n5 {' d
    <w^2 - 3, 1>
. ]2 |7 A' ^* Z. f% c+ R4 ?]* o5 ?3 }8 z& I- z  d( _/ P
5
. n1 `6 N' w; i; N1 K$ q6 y; k: R; w1/2*(-Q5.1 + 1)6 v+ a, U5 b: B+ j3 l
-$.2 + 1+ t/ l* V* H) Y: p
5
% v% n" G- J# n+ ]& m; PQ5.18 c* W4 f4 X1 [5 n9 O9 F' N
$.2, F* ^/ ~9 \" ^9 l
1
  _/ g' a9 Z+ QAbelian Group of order 1
* n* T5 N' [2 W7 S& J3 w( O2 i! iMapping from: Abelian Group of order 1 to Set of ideals of M. J* m+ ?3 X  z4 U  Z
Abelian Group of order 1
9 d0 r5 [. g$ y" J3 s0 @& PMapping from: Abelian Group of order 1 to Set of ideals of M
5 Z5 r3 p- A+ m6 c+ d1 E/ N% H1
# d) [& j8 J- {/ E. G( L% L1* d) r# x$ b$ g8 ~7 Y: }9 T2 Q8 M
Abelian Group of order 1
1 T' w- }7 Q) C5 }9 S2 n- t& }Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
; ~" T0 ~) f& H4 K5 [  g8 ]inverse]
0 ~1 i! E% c! G) N+ M0 N1 i5 A' d' u1
/ c- D. u# D8 S+ O7 X% j% o9 R7 L/ SAbelian Group of order 1, X6 [* Q6 Y2 A' o
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
& \1 V: A- }0 v5 given by a rule [no inverse]3 m$ q$ o8 T$ I& s3 r
Abelian Group of order 1
% {- @5 M7 [0 a8 j/ tMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
, K  \% T; ?" J* d' s7 L; o5 given by a rule [no inverse]: y0 V, B+ a8 s
true [ 1/2*(Q5.1 + 5) ]+ T( m7 [2 z: \
true [ -2*$.2 + 1 ]
* U) ~# w0 c% M" \# @" l3 @1 g5 J4 \: h  l# h

$ j! D7 L& g8 b+ E6 p: J' T
! T, \1 {$ x, j* E$ |, F3 u# D$ b" R4 j; {: x8 l$ u+ i+ B

0 R/ z' v( W# U3 N8 F5 J2 `) m+ [1 a. N( A
) K' C- T7 }8 J+ R/ r8 \$ {
. F! d# F+ ^9 n

8 m  h7 E* H! b& ]# K3 }
1 V9 n6 n6 d4 I- z* r2 Z7 B
8 c' S* S. J0 ~* D" o% m2 z: O==============
  ~6 p/ S; L; V! X$ x( {' d# }& \, J# E! Z' h5 V, m3 k8 b' {
Q5:=QuadraticField(50) ;9 S: K' [+ C; |$ ~; d
Q5;
; Z& O4 f. K# j
; `) Q, ?3 r9 LQ<w> :=PolynomialRing(Q5);Q;
4 d: a+ b6 W9 i3 X  iEquationOrder(Q5);: p' q# d+ M- k7 e
M:=MaximalOrder(Q5) ;
6 J. H, W3 h. a% H" @8 x$ mM;' s; e9 t4 g2 h. `# [
NumberField(M);5 `: w0 f; U- K) Q4 [
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;
* @1 C% ~: C+ y/ l6 aIsQuadratic(Q5);' B, K) ^# x7 ^
IsQuadratic(S1);
8 P) c6 k; K, T1 w1 F+ p* M% iIsQuadratic(S4);
+ O0 I( C8 f% g! U  ?( D( mIsQuadratic(S25);
$ J# g  x* `8 l# O$ Q7 _* kIsQuadratic(S625888888);5 V/ ^' A* }7 D3 D6 h% u
Factorization(w^2-50);  1 r* {& H- t9 L$ m5 U
Discriminant(Q5) ;7 [+ r0 T$ t2 L7 @  t6 m; _
FundamentalUnit(Q5) ;
0 r6 G3 T# H: PFundamentalUnit(M);6 a7 t! d2 l" c4 u: O2 o! I9 n
Conductor(Q5) ;
. N6 W5 H: T8 S: |' [
: e% b/ l# O" ZName(M, 50);
/ y# d9 z5 B3 P' ?, u3 MConductor(M);
5 q9 S; O5 n, R+ T, D+ n: J8 ]ClassGroup(Q5) ;
: D  ]# p; h3 V. K3 N8 bClassGroup(M);
7 R. ~  P/ x% z- C( ^! z2 `! n% K8 BClassNumber(Q5) ;
. X4 J. _* j. AClassNumber(M) ;2 O4 C" S1 u$ j) E2 o
PicardGroup(M) ;) G5 N8 K4 t# G
PicardNumber(M) ;
" J% |" k3 A! n' ^7 M5 z1 f4 a
4 U4 Q2 c% f+ ?2 ^6 T0 E" OQuadraticClassGroupTwoPart(Q5);
% Q. }! C+ u( }! o% FQuadraticClassGroupTwoPart(M);7 Y6 v/ `4 c9 r! F7 h
NormEquation(Q5, 50) ;
9 ^# @7 G9 L1 S) t0 i) ^% `) H. ?NormEquation(M, 50) ;
. I" }8 F- u2 }4 ]( h- X) Z. F" t/ q+ N0 ?, B9 `! O! g
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field) O% d; Q, v' O2 c
Univariate Polynomial Ring in w over Q5# m/ D5 s4 |$ w3 V- x6 r
Equation Order of conductor 1 in Q5! w, ^$ i: U0 F5 Q* l
Maximal Equation Order of Q5
. _0 A/ }$ P9 \! V5 ~Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field* N  c4 {( |( ?1 u) }9 ?. n- @
Order of conductor 625888888 in Q5
/ i& M$ H6 C- A. M4 rtrue Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field$ I0 ~4 X; z2 G4 t  C: C
true Maximal Equation Order of Q5
" h9 t3 F& @  A5 O( etrue Order of conductor 1 in Q5
- B* y$ F2 ^- z1 V% Mtrue Order of conductor 1 in Q5; @7 g- J7 J: K  t( J1 U/ F
true Order of conductor 1 in Q56 R- T7 C- g- Q9 b7 W" f6 x  ~
[
4 [& ?6 S  ?. u- A4 w5 ]8 O    <w - 5*Q5.1, 1>,
) i( ~& y2 g! p( r    <w + 5*Q5.1, 1># Q; Q' o  p5 }. j4 ]3 B& ?( V
]+ H5 n# u, X# A
88 A, ^& w0 ?. a/ |! o# I, D- |
Q5.1 + 1
1 [% O. @% \1 _7 n$.2 + 1
2 g; K3 s% J8 ^3 o. J8( B% @+ G3 T  ]. L$ Q* r

% D: t: S* w8 T9 u, M>> Name(M, 50);
% O  u; [- F" u/ g) W. K# d. Q       ^
$ y' ~6 @' x/ d! L+ Q2 JRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]. E: I5 Y. W* J* g
/ U4 i- j# u; C) y% g8 S" i# {
1
; B4 ?! C+ ?  X2 DAbelian Group of order 1- j8 c, m" _& V
Mapping from: Abelian Group of order 1 to Set of ideals of M5 F: O, {7 E2 {" M) C! J
Abelian Group of order 1
" B5 n9 J% [' P7 IMapping from: Abelian Group of order 1 to Set of ideals of M) a( S' ?4 R1 S; P- z( F7 b4 L! `
1
+ `7 O9 t' J& E( G* o17 ~0 s$ p4 _+ t! z1 u
Abelian Group of order 1
' t: w: ~9 D/ n- XMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no* @- l. u8 q4 g" X
inverse]# K) _, c; ~4 E  l  Z
1
' ^2 Y6 }+ E& ]4 c/ o3 rAbelian Group of order 1
6 J; l3 S& {" ?. _$ i  A, UMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant- ~1 O; S4 l7 l' i" i0 c6 f
8 given by a rule [no inverse]. @+ V2 r# i1 k4 ^
Abelian Group of order 1
3 Z2 n9 `! k. c% W; O, h1 k  JMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
! P/ n* k% L* w  V8 given by a rule [no inverse]5 I8 \. Y1 r9 z4 M
true [ 5*Q5.1 + 10 ]! N8 Y9 w* O  Q6 C" M3 G% a
true [ -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
: g9 b& T+ F& U( C9 X) e; J) f/ ^8 C: q+ v' U! I9 P1 v/ z  |: J
基本单位计算fundamentalunit ：1 O9 z2 ?# U! f3 A. P
5 mod4 =1                                              50 mod 4=2  c- N4 S- y; \9 z5 l

: [. C8 D" h0 N" I- ~ x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
$ y9 B; M9 W9 k* Z% E x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.: @7 h( h# K) \) H+ q' h
, V+ B1 i/ D; q

% t. f1 I7 f( F  \8 {3 G最小整解（±2,±1）                              最小整解（±7,±1）
+ _0 ~) O5 r3 ?0 q/ |: R& j% i                                                             ±7 MOD2=1. S; m0 n( W3 X( D+ e4 j4 c! W1 E4 p

% y  U( H& X) j3 t2 `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 # Q6 ^5 _4 h5 a  h8 l  x
基本单位fundamentalunit ：/ W; R; D2 A: ~' ~
5 mod4 =1                              50 mod 4=2

$ _9 I4 z7 y5 @% j4 U6 u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 % Q/ y1 e3 r0 n2 f, ?& {
# f7 |6 W8 T. o" ~
判别式计算Discriminant
( B2 `  E+ |2 v- j  h" q! p
; k# l& u& Q9 }1 m  V5 ~4 t5MOD 4=1
7 W- _- p4 x( X# }3 _) f: [1 m) \, O/ P# P+ |& {% T
（1+1）/2=1          （1-1）/2=0+ T4 T1 N' k+ ^0 v6 ]' l% Y5 M: V

1 a( r. v3 M4 U- iD=5
; K; J3 U- w1 L1 c# q8 O+ z6 s9 w* V% \2 T+ p- F
7 |) F3 w; x  K7 t% y
50MOD 4=2
* g; y: |0 `; hD=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 2 b8 E, s' Z' h7 _% @

5 B. @: V& y, [- D' U% ~# r分圆多项式总是原多项式因子：7 H9 S- d* M3 R. [
C:=CyclotomicField(5);C;( g, R9 F8 N3 A
CyclotomicPolynomial(5);

6 G: ]/ C! j3 b; q6 I
. v5 b! w" x4 E* @' a. j分圆域：2 j+ {% Y  \  l
分圆域：123
2 |/ J! g# X2 R# n! B
9 [3 {: q% v  i7 z, Z6 ~7 cR.<x> = Q[]
' D6 l% @6 n2 ZF8 = factor(x^8 - 1)
) E" T& l: H/ ]$ _3 G# AF8! g4 K; M, x9 Z$ T& ]  X! U
$ a7 K2 z2 c0 }5 @( `9 ^
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1) . q2 H4 d4 p6 Y3 a3 s( w# O

+ S* r7 A( }+ Y9 i7 r2 C7 L* ]% ]+ PQ<x> := QuadraticField(8);Q;5 h) o6 ^7 H, D* Y) k+ p3 x+ J
C:=CyclotomicField(8);C;$ W3 T! v4 F8 _
FF:=CyclotomicPolynomial(8);FF;& T8 F; P! v0 D
' i5 P6 J/ T! R9 o* ]
F := QuadraticField(8);
, k  i' F( }$ t& Y- d0 NF;
; J4 @  F7 Z; m6 t5 H/ K% uD:=Factorization(FF) ;D;
7 p) f5 e, O! PQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field5 q# K4 Y) O$ G5 X) ^' @
Cyclotomic Field of order 8 and degree 42 ^. k3 k7 a. K7 ^  ]& C$ V
$.1^4 + 1: ^, b" n- S; [, J" X9 ?( j
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field' Q8 U4 N$ [+ i9 v% z/ e" O  w) t2 T
[
9 B: Q& n% n. u9 V1 g% d+ e3 \+ X' n" W    <$.1^4 + 1, 1>7 G# f; k7 q8 B2 I1 d
]
& e% \% T6 t6 A, y) b9 o& F: d% T3 h0 R  k1 E
R.<x> = QQ[]& g0 E3 o1 V  B0 S' E
F6 = factor(x^6 - 1)8 G5 h& s2 E% S: ]( E2 l% L
F6
8 T0 i% f5 x' d/ Q3 j% W  Q& E; j7 Q- R5 P3 ~4 {4 X
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
+ k) S; A  v' ?3 H5 N+ Q3 a9 H" b1 Z; E
Q<x> := QuadraticField(6);Q;
6 a1 N( Q+ U8 P8 u7 _% e# ^C:=CyclotomicField(6);C;
% t8 D5 |: R! y% v+ ^FF:=CyclotomicPolynomial(6);FF;
; c" n8 c& C' E
" [; B$ @/ Y7 R1 oF := QuadraticField(6);. z/ o- `$ w6 @
F;! b+ K" a; z3 l: X( S) [" G( \
D:=Factorization(FF) ;D;
: f) H7 q5 e7 o5 N" UQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field6 J# p8 N, R, q
Cyclotomic Field of order 6 and degree 2" o1 Z% `1 o: m: v
$.1^2 - $.1 + 1
- y% q4 z3 `8 O/ i4 z5 M0 y6 }8 EQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field4 b* ~5 u4 e( ~' Q" w
[* a, [2 ?& e/ T1 o5 f9 h( h
    <$.1^2 - $.1 + 1, 1>
! W% e( _7 D/ ?% v  Z/ `% G/ Z5 y]1 B' }; V' e- K8 |
" K9 _1 `0 b3 i8 Z# S
R.<x> = QQ[]
* V3 B- G( Z' MF5 = factor(x^10 - 1)
4 E1 |  X& p9 O1 A) ~# B( s' L$ n, x$ zF5
6 v0 q8 _9 d/ d3 Y; ]5 a* ?: j- [(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +0 E& I- N; \+ w$ x; ?- ?! S
1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
! c3 ^8 B6 u# B3 f: K6 z" |" g' w' J: ^- `# ^* p  ?. n
Q<x> := QuadraticField(10);Q;! l9 I% \0 j$ N8 _; Q1 a
C:=CyclotomicField(10);C;
" O: L) n0 ?( ?6 o5 r6 E. z7 hFF:=CyclotomicPolynomial(10);FF;6 M' b7 ~) ?0 ~
3 }) g  i+ s0 E
F := QuadraticField(10);0 Q- ?7 T8 d! K9 d, f. G1 U4 B/ ]
F;. C8 N+ @0 F; X
D:=Factorization(FF) ;D;
. n- X& W) V* G1 h5 ZQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field0 O" R; K' M: B! n; x
Cyclotomic Field of order 10 and degree 4, n. O7 J- ~( A$ ^5 T# d! e# o
$.1^4 - $.1^3 + $.1^2 - $.1 + 17 }! ^2 S! Q0 v' P& m0 h
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field/ V. [0 Z* S( x1 R# F
[
0 H; O0 j8 ^' }1 M" x  p' L7 A2 a    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
2 h' y: M3 t5 J. j" m* S. U]



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