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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 , s7 F, {7 q) T' Z) t* o
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Q5:=QuadraticField(5) ;7 M5 m; H" A/ E( I
Q5;* J( `0 I6 B: w- g8 |
Q<w> :=PolynomialRing(Q5);Q;+ L9 R3 h+ a1 q1 ]& M

+ O$ m/ j6 K7 d$ g" ]EquationOrder(Q5);
& W) {# `  ]2 B' {2 i- A' tM:=MaximalOrder(Q5) ;# D2 I; _9 g' \9 w- K' x& p' E
M;+ e9 E! a, N) ]# _" X
NumberField(M);) ^' @6 ?) O% d1 i3 S$ R
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;" s! f8 y& C% T/ [8 @) [6 h
IsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);# ^" @% x, F9 _& M& H
Factorization(w^2-3);2 M+ b/ g/ ?& T" N) p$ n( A! D
Discriminant(Q5) ;$ k* l3 N. ]& D
FundamentalUnit(Q5) ;
! S9 |+ C* Q( G" m! fFundamentalUnit(M);
) H8 O* V- P; F$ J1 QConductor(Q5) ;, o: z% r! H1 F3 [% P( |# L- h
Name(Q5, 1);
$ E. j' L: v( ZName(M, 1);
9 W" D; _, m: D3 fConductor(M);
" b: M- H  v! v! kClassGroup(Q5) ;. V! L) b  x7 q/ N* d! c/ o
ClassGroup(M);
& W! ]/ O! ^9 u6 O$ EClassNumber(Q5) ;
' A3 x1 [) a& Z8 w/ M1 j. eClassNumber(M) ;
0 G# x& w$ N# G# N6 F( N6 G2 Z
. W: V1 b. {$ b* W/ c. f9 MPicardGroup(M) ;/ @" R) N3 @4 _
PicardNumber(M) ;
- I) G8 G, s' K  {$ d3 s3 Z# t5 O) F* b4 Z5 T
: i' B5 W# P$ y& p
QuadraticClassGroupTwoPart(Q5);
+ ~: Z8 E( C: e7 M+ R' q( kQuadraticClassGroupTwoPart(M);
  F4 r( `  S: R* @* U, k8 ?
, T/ [% Q- I8 l6 l! j$ w
: ~; m% u- s5 j, ?. _3 i3 FNormEquation(Q5, 5) ;& h& ?/ T/ V( R) H8 O- z" P/ [7 m
NormEquation(M, 5) ;% f; M0 w7 d/ `* V4 |

6 |) ^9 }9 m/ m# n) j# w) p+ S+ @
; t7 k2 Q) `7 _- b) h/ RQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field# D5 f- @7 B- t
Univariate Polynomial Ring in w over Q54 J5 y) @. a% ~( ~5 f# ~8 L8 q
Equation Order of conductor 2 in Q5- R9 T( r+ G6 ]' c
Maximal Order of Q5" Z  i; k( I' z2 U+ ~; [1 K
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field* U2 @# ]3 g' I* X8 r
Order of conductor 625888888 in Q5
6 a' i- m# f- m0 @; g/ y" |# Btrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
# C6 w$ Z" c; ~  L" Htrue Maximal Order of Q57 `) j/ F  ?" P! @
true Order of conductor 16 in Q5
7 {3 [5 F- n) y! p- btrue Order of conductor 625 in Q5" ?' C6 Q6 L  C- v
true Order of conductor 391736900121876544 in Q5
+ }  C5 r0 {5 G1 ?[% t* v* H$ `, S) m9 c/ Q# B& l
    <w^2 - 3, 1>
, J/ Y4 g  L2 L+ N, W. \]
3 z4 P# t# |8 v9 Q7 |8 B* }5
, Z" c# E0 w( ]7 k1/2*(-Q5.1 + 1)
3 b& b7 e2 e4 W/ k! G-$.2 + 15 w0 W1 b' h% }# m; O/ r
5
/ ^2 \% r; B. D7 B2 [1 mQ5.1; y+ {1 y* n$ |6 L2 e3 `
$.26 ~1 S1 N4 k) t5 r9 f& S6 R$ {% {1 E
1
0 N! u+ d4 D9 [6 |2 @7 L0 sAbelian Group of order 1
6 `5 I. F' {$ }1 ]- KMapping from: Abelian Group of order 1 to Set of ideals of M
' ], ?8 a6 ?; v9 hAbelian Group of order 12 N% f7 c+ H# r
Mapping from: Abelian Group of order 1 to Set of ideals of M
2 O0 G0 Q' v7 R: b8 t! G1
/ ?9 G% n8 d, e0 A1
8 |& K8 J! J4 f3 w" N9 @! \Abelian Group of order 1! R* L' e8 H9 r1 {: [& b
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no( E. O8 r: Y  f+ F! p
inverse]
  m4 e6 t4 J3 }1
# w' H' t9 i/ t$ e4 cAbelian Group of order 1+ `" u* a& F. x" g
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
; Y; J+ U% l' q* C, r; o" C' v- L6 R+ Y5 given by a rule [no inverse]9 ]& A. G3 G6 b
Abelian Group of order 1" j2 Z) N9 K7 Z6 j5 M9 @( Y
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant- M6 T- M, d" Q
5 given by a rule [no inverse], r) f" k% p, v$ i2 I9 r
true [ 1/2*(Q5.1 + 5) ]9 b+ s' ]5 _! y8 |8 l; G; j& M
true [ -2*$.2 + 1 ]2 ?5 a% k8 P0 x+ _3 |8 F2 e) M

0 l% x% S0 i2 Y4 S, _
: y( g, M" k" E% j- {; v9 M) {" w1 b) G5 C/ v4 \6 A+ U
" @- h# q# s  R. \, z# \5 U

4 T; T" H1 Y6 L% E7 }3 t+ ~: j" ]. P2 b; h1 v
7 r; `' p1 I( x- l/ b
, t+ k; c! ~5 W) W
) V" N) r8 z9 C7 N7 T
: s( o9 L2 n0 p/ v& n+ g8 _

4 b! r8 Y( B, R* y) _. i7 L2 s% l==============
& ^8 c0 L2 q6 E6 v* A1 {
/ j9 n' Y+ Y! e, P/ PQ5:=QuadraticField(50) ;
' {& n3 G6 ?( q8 G7 X. V- _3 }% VQ5;7 J$ L. m% A+ n+ B5 `! C, S

0 M0 G& b0 d5 J# M3 mQ<w> :=PolynomialRing(Q5);Q;
/ r& a3 f1 l8 V$ ~# A- b! A' S0 s2 gEquationOrder(Q5);
. W& |" T6 F; ~9 mM:=MaximalOrder(Q5) ;) i& M5 n( {/ |, ^$ x) @. e
M;
! }) H% L/ m- z9 A# E& @1 bNumberField(M);
$ a9 r/ w7 s: u+ m8 i: p! XS1:=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;/ M3 O+ g' T4 l) c
IsQuadratic(Q5);
  }3 r0 k+ Y* q# U6 ]IsQuadratic(S1);2 d+ H3 m% x5 M
IsQuadratic(S4);! F+ h$ K' i" J/ I6 h
IsQuadratic(S25);
) o  E; t; E/ ]  B! o2 J3 vIsQuadratic(S625888888);3 y( L( F. b- t; v9 V
Factorization(w^2-50);  2 P/ ]) y9 h% c7 y7 m% H1 d
Discriminant(Q5) ;; J, B* \( v. w! K; G0 X0 b( ^1 Q
FundamentalUnit(Q5) ;& Z. W! z0 w; K- o
FundamentalUnit(M);* ]3 y1 i  ]) h4 q; O
Conductor(Q5) ;
' C: f8 {2 s9 Y1 E0 u
: N. ~; w  b% J0 d2 MName(M, 50);
. W8 x& t  z; C+ Q0 u2 lConductor(M);
( I; p1 p9 ~1 g( r6 i6 d) jClassGroup(Q5) ;
2 d3 D$ a3 l* P0 R) G2 `. GClassGroup(M);
' v$ {6 _1 d7 G& i! wClassNumber(Q5) ;( y6 L$ E: B# ^6 o+ x: Q/ @  O
ClassNumber(M) ;) I0 K2 k6 m0 l: W0 ]5 a7 j
PicardGroup(M) ;
/ S5 r* j; j! p2 d+ UPicardNumber(M) ;* g% ^# A% l" K6 l

3 I# C9 ?( D9 U6 l4 l: ZQuadraticClassGroupTwoPart(Q5);( I/ N5 l8 @8 G
QuadraticClassGroupTwoPart(M);& T7 F3 L4 [* |! ]/ z
NormEquation(Q5, 50) ;
6 B5 e: W+ b8 Q( yNormEquation(M, 50) ;7 K- ^. m6 a/ I
/ R2 x& u  T, }$ F% G/ `  w' j8 K
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field, W# ^" f3 v5 E/ q
Univariate Polynomial Ring in w over Q5. p- d. {* t4 J5 \
Equation Order of conductor 1 in Q5
" U- b8 {$ K8 k. S3 _) ?Maximal Equation Order of Q50 d: t3 `# x- c) c5 ~
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
; l! M/ _! e1 P' ~) M% YOrder of conductor 625888888 in Q5" l! m2 p- d- D
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field* o+ a9 K+ c% P6 z2 v
true Maximal Equation Order of Q5  P6 Y' H/ I0 s; `
true Order of conductor 1 in Q5+ u% j7 @- e$ a. P; P) R
true Order of conductor 1 in Q50 y* A8 T8 [6 J/ M2 `: {/ O3 w
true Order of conductor 1 in Q5* t# S, N; X. [, m3 s" l- B
[
; l' ]* X! q* L/ k" \7 E& S; S$ s    <w - 5*Q5.1, 1>,
" G9 w# @0 H0 {6 W; y* D9 _7 z6 L; c    <w + 5*Q5.1, 1>
) N; Y& k5 E* q/ l]: P7 D8 ~) J) V3 \% P; a
86 N/ j/ i/ x7 D) Y
Q5.1 + 1
! }) \7 M( Y9 M$.2 + 10 S- G2 [# R4 `7 y- N
82 a4 w/ X7 v7 e: d; T3 k/ J
  B( v0 ]# V: I/ l6 p8 }
>> Name(M, 50);
; ~4 I: o( g: H  ]/ ]7 Z       ^
" N6 Y& Q5 ~( A- l% P8 \' {  g, GRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]/ ~# |* \5 O1 [% ~

% W- e* z! z4 R4 f1% ]( _; O! P1 x* t
Abelian Group of order 10 ~1 \5 D- ]+ w( R+ d
Mapping from: Abelian Group of order 1 to Set of ideals of M0 C: P" d0 M: o0 H5 O
Abelian Group of order 1
# @& l, T3 z! Q" ^1 gMapping from: Abelian Group of order 1 to Set of ideals of M
; X% |. Z6 b' q: p& R6 |1 |1+ k6 D0 Y# ^; ^8 d) V- u" @; C
1$ k" r5 w! l. y6 T9 I" y& r0 N/ T
Abelian Group of order 1
+ v/ L" D& B* [" A8 n5 ]Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no3 j+ M, h1 L( U0 z5 P
inverse]
4 f! s  F" U# L: v1. A2 ?& |$ O$ L' n+ K1 {
Abelian Group of order 1
3 p  N* x' r# B# H0 GMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant+ M4 O7 Z4 x6 j) ^3 Y: N
8 given by a rule [no inverse]
) Z. D% z5 ~& H2 N# U, o/ `Abelian Group of order 1
! K: _' q! Z4 c1 CMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant$ ~6 S& }# \7 D2 n+ c! {
8 given by a rule [no inverse]
# c8 i' O# s$ g* L' Ktrue [ 5*Q5.1 + 10 ]' Q4 l0 K- w% A5 @9 U
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
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6 \; ~. @% F0 ?  }1 b' G基本单位计算fundamentalunit ：$ G2 j. a% o% ~& N8 K5 n
5 mod4 =1                                              50 mod 4=2* e( v- J* }$ ?" c2 e- v

4 S& `6 [* g+ r, ?$ a8 t; F x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
- [7 ~% _1 N9 q8 b: F6 d x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.* O; v+ X/ m* L
& P1 l  M( t# b; \* i1 ], i

; t$ @* {5 x) D) D9 S$ d8 G0 [最小整解（±2,±1）                              最小整解（±7,±1）8 y3 E1 ^% u: X2 b8 m6 w, I$ X6 P
                                                             ±7 MOD2=1! N2 B5 k0 V! v5 \) j, k0 u! t  O8 F

' c/ B0 X# Z, |/ {) s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 x, Q9 `! K# O* R) f' h
基本单位fundamentalunit ：. S1 S( y4 ^; B3 m% z# g
5 mod4 =1                              50 mod 4=2
0 r" a7 e9 B" r
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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判别式计算Discriminant
; ~/ w0 R* \! X* F: n. d% w
5 Q/ a6 a$ R* n. c, m7 h4 D7 H5MOD 4=1 # h; k) ^+ a! R" X

' D0 R/ }  R" g- O6 c! P3 T( G（1+1）/2=1          （1-1）/2=01 L: Q/ J/ n2 t; `7 ~% B# e

) l9 e  H" C- \D=53 z1 ^$ y* r4 D4 _$ i0 {
# \) d: |+ X1 L
! k0 Y. K& Q9 Y( Q' z
50MOD 4=2
; l  {( I! X0 L* \D=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
5 f6 j9 f2 q3 D
( M7 O! K* p* D7 e2 ^+ i: i分圆多项式总是原多项式因子：! a3 J. p% @: M
C:=CyclotomicField(5);C;4 `+ X6 A2 l7 H( I6 B- U+ U
CyclotomicPolynomial(5);

* }$ t1 e; z7 V* v. f9 v3 X' o+ M
# S) }/ O  j/ r+ z" Y分圆域：
1 g% J8 k. A. C( O分圆域：123
4 v$ D8 V" D/ }9 z/ ?" v# O% u  n* u4 t
R.<x> = Q[]# a& }1 ~/ f+ S: {: F
F8 = factor(x^8 - 1)
) L  A7 S4 L. cF8
! E. M7 a4 |1 f& n+ Z  d
7 ~" l9 J1 F4 ](x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1) / b/ P# ?3 ?5 j6 n
% e2 P6 v8 y3 D
Q<x> := QuadraticField(8);Q;' p7 c8 z. B3 {  `3 G8 f! w# r
C:=CyclotomicField(8);C;! N8 l- N9 L8 q+ Y
FF:=CyclotomicPolynomial(8);FF;
0 L" Q' E) X7 b% U, ~5 h, I# y) Y" n+ \
F := QuadraticField(8);4 \# {: B; l6 \6 F" b3 ]: Y
F;6 g3 t6 O! N5 V) V' Q
D:=Factorization(FF) ;D;% o; I8 b8 u  E3 H+ b3 h+ I
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
4 S) G2 P- Q. h0 b" ZCyclotomic Field of order 8 and degree 4
9 D0 K  f( u# v* D6 N$.1^4 + 1! ~' |* Y- A* a" T- d) e  c9 Y
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
& A% B1 |; |1 {: W8 F[
+ }  h# Y" t" X. ?    <$.1^4 + 1, 1>
& q  B+ y, ~1 `: O, O], M; }4 e3 P3 v/ L

7 H& R, X5 @' N% W3 CR.<x> = QQ[]4 t0 G$ d- G% }. M7 v( H: z8 r
F6 = factor(x^6 - 1)
2 U' i! R- [+ ^  r4 N) z! L4 JF64 J: g6 u5 r0 F- Z9 V( d5 @* R

& N: C( E$ k* k8 j% n- r0 g2 k(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
9 g' a" E3 _" t3 `; R" d7 _- i* d3 G- x; Y* ]) c" x  H+ l
Q<x> := QuadraticField(6);Q;
+ u3 h" i5 b7 W$ z) w+ r+ w. gC:=CyclotomicField(6);C;. f, a( o1 G* X. I% F
FF:=CyclotomicPolynomial(6);FF;8 J$ q' Z% a- @, ]

( l$ O! V! ^& K, Y$ F3 A$ uF := QuadraticField(6);* f/ c# q) M: D
F;; I7 l) d0 ]7 V
D:=Factorization(FF) ;D;
0 k& B# p2 n% S4 lQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field( y" c. ^2 J% I% C+ G
Cyclotomic Field of order 6 and degree 2& j( ~; v9 Q: O; d& B! ~0 M9 I
$.1^2 - $.1 + 15 Z$ R3 d# h: U3 X+ ?$ w" V
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
2 y  ?5 {4 Z$ x: x% e$ H5 c- n[
. d" K3 b1 q8 p9 ?+ M    <$.1^2 - $.1 + 1, 1>4 B2 P3 P7 o% L  D5 Y1 L' l
]% g* q' F. N3 U/ O  f2 p

. b; {. `5 J4 o# rR.<x> = QQ[]
) a2 k7 ~1 H- }7 I* m/ SF5 = factor(x^10 - 1)! b; Y( z+ `% n" @, W: P- {+ H4 A
F54 U4 K- m: l* A) u9 \8 d
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +$ f4 |1 \6 z# {' K  F" \5 b4 u! J
1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)- S. C7 e; Y  u
: \( m" b* `2 t& _) y+ i
Q<x> := QuadraticField(10);Q;3 E6 l/ R! I" ~( i( r
C:=CyclotomicField(10);C;
) K. H# H' X4 z5 y; P! K: SFF:=CyclotomicPolynomial(10);FF;+ D0 W% `# g- U/ d: h; @

, ?7 `' ^" _% j2 _9 JF := QuadraticField(10);
* H& P2 Z5 W0 m' f: \- JF;
  p0 O% l0 v, J$ [D:=Factorization(FF) ;D;1 c3 Y% P9 j1 n3 L  U# ^7 P
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field3 U  ?3 K/ q8 ?  v' A9 E
Cyclotomic Field of order 10 and degree 4: d; m* E2 g: ]" ?3 f# b! `9 a
$.1^4 - $.1^3 + $.1^2 - $.1 + 11 q# U8 {- s, z
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
$ f9 O+ v: K* g* r- ?[
" p& w# V0 @$ c6 o) L9 }. Q    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
) j& E2 b5 E3 ]  n8 E: F* _/ \; n$ G]



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