数学建模社区-数学中国

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
* x3 s* Z; f& x! x/ Y; k# h
! W. [3 E9 a! {8 Q( ~  rQ5:=QuadraticField(5) ;
# D) U; l% k% O; P5 VQ5;
6 o, @0 g/ _: u* g( D5 @$ LQ<w> :=PolynomialRing(Q5);Q;2 I5 N& J: \, o3 s3 F; B
1 |+ f0 ]  H% D/ _/ {
EquationOrder(Q5);
: n( V: t9 {% Q2 TM:=MaximalOrder(Q5) ;( L. j) z' S! c2 }4 ^
M;
. _" ~( K2 d' VNumberField(M);/ S0 p. g' y' r6 p- k5 n" J" 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;: ]0 b" C1 G- L$ ]
IsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);. I' h8 {  n. |0 U! B) R' r- w' W
Factorization(w^2-3);  Q0 s% R5 t9 g: O) Y+ n
Discriminant(Q5) ;! }+ ^5 X# {- z# x! j; g7 q8 R
FundamentalUnit(Q5) ;  g! X2 E. |3 N4 c! N2 m& u8 l
FundamentalUnit(M);
6 N! w; P$ k* s/ J) o2 |- _Conductor(Q5) ;
& ^+ b* X( k4 ^& s  |" yName(Q5, 1);! D0 \% @% A$ x3 c
Name(M, 1);
, T1 U, W4 x8 |9 ?Conductor(M);) O; [* Q6 f* W, q
ClassGroup(Q5) ;
0 W4 Z# ^& u' v2 TClassGroup(M);
4 D1 d4 P! S; g4 \$ a9 C7 R! x+ |ClassNumber(Q5) ;
# y, x8 {  y, v1 n7 xClassNumber(M) ;
: r: Y) ]: L+ Q) U
( a; k, z7 ~# ]! D& H' a( lPicardGroup(M) ;
# N; g% J1 G1 W" c9 O- sPicardNumber(M) ;* u* B: l0 P5 Q7 L7 }" e6 n2 t; B9 w
' F3 l# Z' R: W/ K8 L

  d& P* x0 m* v7 n# C1 cQuadraticClassGroupTwoPart(Q5);
9 V6 m0 m% g; j2 H! u0 JQuadraticClassGroupTwoPart(M);" u: J% n2 M6 _' A8 g, H

: w% g5 u: B/ C# o. ]9 ?% c; F( x) [) v8 p% C( Y
NormEquation(Q5, 5) ;
' b5 C3 u  X' A7 |. Z, j1 YNormEquation(M, 5) ;
; y$ Y8 @% w4 f2 u% K# e0 H
# r! n5 ~1 O! W7 a  T  k; g6 y; g7 M, D, g% C
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field3 E9 _, Q# Z! o6 \, v  W# A
Univariate Polynomial Ring in w over Q5
* ~3 s9 j9 @  q) o: @Equation Order of conductor 2 in Q5
7 f7 G9 u9 x/ I' }+ U, K3 z: CMaximal Order of Q5
3 R$ {( v% j$ wQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
2 {. f6 }3 n$ @4 M. F  HOrder of conductor 625888888 in Q52 |5 f1 o/ I6 U  ?
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
& R5 z5 r* `' c: u0 D0 `true Maximal Order of Q5+ J4 e& D, n3 a$ f) c# c1 j" s
true Order of conductor 16 in Q5" c( N7 W8 r, W4 g+ i6 `8 p" K
true Order of conductor 625 in Q5" x+ n' b: }- V( R6 T! E/ G( m
true Order of conductor 391736900121876544 in Q5: |; \7 a4 M/ n4 j, \9 H
[
6 G; A% d4 r* @/ m8 ?, P    <w^2 - 3, 1>
, g1 d6 W* ~" E- K1 J# _4 z]
3 E: Q3 _9 y1 S5
! H' G# g% t# O, }' z9 X1/2*(-Q5.1 + 1)4 Q- i6 T$ ?# w/ x
-$.2 + 1& q1 _* P5 X& q; D+ l
5
. `1 v# D$ ~1 ^; ~! iQ5.1& @/ Y+ O) h: `+ @- {4 \1 o
$.26 p* h2 k% J5 \. ~- P3 D6 E
1
" r; G) t& j: T/ X/ eAbelian Group of order 15 y8 G9 e- L( G
Mapping from: Abelian Group of order 1 to Set of ideals of M
6 d" O8 H( j- x5 WAbelian Group of order 14 I4 q5 K, c' v. e
Mapping from: Abelian Group of order 1 to Set of ideals of M
3 @5 k4 F. ^. J: i, W1
6 s/ `- O" E" u1) I) Y0 }, G" i/ U# P. B1 [0 F
Abelian Group of order 1
. r& {, d/ \, M  |3 EMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no( c) H- i% c; C
inverse]
% ^2 }) w" @9 p# W) I" F6 }1
; V9 }4 J" ^  e* i6 h# x% ~2 pAbelian Group of order 1' |0 _5 l: N' {. t* Z& K
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
- m2 _4 k# ~4 @! \7 ]) |5 given by a rule [no inverse]
" e* E. M4 J7 M+ R( Z: j0 V9 q( tAbelian Group of order 1/ A6 q2 N2 f! I5 u+ T. ^
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
. O0 |' }, r, e  A0 A% I5 given by a rule [no inverse]
+ ^5 ]8 b6 u4 [4 xtrue [ 1/2*(Q5.1 + 5) ]
# l7 w, R# v% q# h+ a# L& M3 W0 vtrue [ -2*$.2 + 1 ]
) g9 U8 R6 Z& g' F% R% f7 Q% S! ]' D4 _' u  `

, @  n) L1 ~& C+ S9 c7 {. b: V1 e+ y
  v. _9 R& O6 [" z( Q9 ]

: \& i1 q* m4 b; w6 _1 a" K5 |% v1 e
! U3 e1 {( j+ }9 T8 x# d9 Y% x: R2 X1 H& `$ U$ A3 F. o

( j, i/ X+ O( f# V! U& ]% y
2 V& b' u# t- ]+ ]6 r$ f0 I3 q9 Y4 D7 m' F; Q! S1 I

! A- M8 J8 h! \+ R4 Y. K==============
7 o6 m5 P. }$ R! n
# I3 ^% Z7 k9 ^Q5:=QuadraticField(50) ;  e9 |! ]# A, e$ b- |+ t
Q5;! ]6 O* r/ S% B0 a. _( o

& Y  ]/ a+ r' M4 W; ?; N9 t- l- oQ<w> :=PolynomialRing(Q5);Q;
! ]" F$ B, v$ a8 l( L7 HEquationOrder(Q5);, s* W1 g. P& f, q* V+ g1 G& f
M:=MaximalOrder(Q5) ;
  h3 c9 X: |# _M;  C& j) A! p4 ]% V/ l
NumberField(M);
& P: Q# ~& R  q1 @5 N% T; {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;/ {5 ~6 O. M4 N. e) l
IsQuadratic(Q5);
! k. W2 \& y* y9 _* pIsQuadratic(S1);
) _, C8 J4 j% u* ]5 P# Q5 A& HIsQuadratic(S4);' I5 W& t( A2 i" \  [! [# Z, ~) s4 ]
IsQuadratic(S25);# _$ X- E7 t$ o# N; e/ q; a' D
IsQuadratic(S625888888);
/ C* Q7 Y' M- J$ G& f3 yFactorization(w^2-50);  
1 y% `6 W$ w( }2 G6 I9 S/ `Discriminant(Q5) ;
! {0 U: b  }- e0 ]% CFundamentalUnit(Q5) ;
6 ?7 o: }7 w% v9 PFundamentalUnit(M);3 x1 l* J3 J' F8 u6 B9 h
Conductor(Q5) ;; X: U  i  `1 @* P" B# S: \
& J- Y3 g: e0 E9 q; B3 s. w! E; K
Name(M, 50);
1 N1 r0 ~  k) T  y9 U9 k  J7 KConductor(M);. i+ ?( R. X/ M7 b/ I) Z
ClassGroup(Q5) ;
4 u6 D5 f% D2 `3 sClassGroup(M);; m0 b" [# n' B% z, C! F2 k
ClassNumber(Q5) ;# Q. G& Q: e" [# X7 B# ]8 J
ClassNumber(M) ;7 J; t& ^! u$ T) l
PicardGroup(M) ;
: }% {( v( e& N1 @% h  y3 J* ZPicardNumber(M) ;. C; Q  V: ^1 G. ~: Q4 [" t

+ R6 m! i0 n- d' N2 w2 bQuadraticClassGroupTwoPart(Q5);4 N! a2 G9 w* J, _3 [. |
QuadraticClassGroupTwoPart(M);. N) F  p$ ~; r- L, {' a
NormEquation(Q5, 50) ;! l. a* |$ f3 a) g3 L! ?
NormEquation(M, 50) ;
) u) L, A$ C; i$ f  R' C
2 b) ^) H2 {- O- rQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
: ]3 P) d# l' ~5 v5 }% g4 mUnivariate Polynomial Ring in w over Q5
6 q$ d4 Y6 O) N" uEquation Order of conductor 1 in Q5
( S- b: l+ d1 X0 E8 wMaximal Equation Order of Q5
9 D: `& q" Q5 ^7 N2 a: V8 YQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field, F0 ~( L! V4 \; x: D/ k
Order of conductor 625888888 in Q5
) o. i$ f  J9 a6 \0 itrue Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
/ y$ B) ~- J- Q% N" t! utrue Maximal Equation Order of Q5% y4 K! \; Q9 s/ o+ o% t. `( a1 v8 i5 g
true Order of conductor 1 in Q5/ f! k! y# c+ K
true Order of conductor 1 in Q5; |3 M# u5 y. F
true Order of conductor 1 in Q5
  I& l6 }/ s  k( B+ s# B[5 b* k0 i( {$ J. k, f
    <w - 5*Q5.1, 1>,2 S5 P( {# L6 [$ U3 B) V& `7 w
    <w + 5*Q5.1, 1>1 q+ e0 b2 {/ X3 v$ {! f
]
8 D0 c9 c. M" z8' ]7 H; d5 F. J
Q5.1 + 1
( q! t2 \) e: L2 E% X- ^; [/ A4 [$.2 + 1
0 F( ~! w: h* c$ r/ {8
: H  R3 x+ ~' R' t: X1 d. X& K0 ~- _( W, G
>> Name(M, 50);: U" u  i- {, b6 o9 z
       ^
3 d* b; f4 T, G6 w3 v- tRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
0 {: W$ R( y& `4 P$ O5 O# {8 D) B! L. }1 u* U- D
1
' b5 L  N2 v) r0 U9 o, sAbelian Group of order 1
  O' c, E( H1 K3 X- \& f* ?+ _Mapping from: Abelian Group of order 1 to Set of ideals of M4 B! f7 m0 z$ F$ f
Abelian Group of order 1
; W( r6 I# {5 _* KMapping from: Abelian Group of order 1 to Set of ideals of M
+ U% g, B6 T7 f' S2 v  z# |9 F1. p" r1 P- M6 C  p( H2 q7 ]8 S
1
* ^. o# ~0 e* U0 FAbelian Group of order 18 ^/ I$ f% W3 h# x+ q  p  m
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no, o& A  [6 b3 S, F4 d, U2 k  d; r2 [3 Y
inverse]
* b: K) D3 J- ?2 r9 i& s8 z9 F1
6 @9 B8 v% F9 ^Abelian Group of order 1
  z- B0 h. b* |/ K! z# iMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant4 Y: ^0 H8 h: ^# f
8 given by a rule [no inverse]6 _8 f4 k4 B& {- J* \. t* q8 `
Abelian Group of order 10 _% A' U* W5 w
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
! j, _9 }. X3 e/ M8 given by a rule [no inverse]$ X+ C; h) U! W! d" m3 b) y1 {8 j
true [ 5*Q5.1 + 10 ]
  h' i' l* `. G# utrue [ -5*$.2 ]
作者: lilianjie    时间: 2012-1-4 18:00
二次域上的分歧理论

1.JPG (177.16 KB, 下载次数: 328)

1.JPG

作者: lilianjie    时间: 2012-1-4 18:31
本帖最后由 lilianjie 于 2012-1-5 12:42 编辑
) b( H0 W( m6 Q4 G1 ]3 m- Z. I- R, j, r
基本单位计算fundamentalunit ：
1 s- m% f, a5 G/ J9 O4 \8 W0 N5 mod4 =1                                              50 mod 4=2
) K2 t* {3 Y. r4 g" g+ n0 N  X$ k8 N: x. j, t
x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.9 q& Q: v$ V& k+ N! ^, r8 {
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.& B$ _% U1 n% C

2 V2 N5 |% @) c+ L/ Q" C; Y& z  K" p7 G7 a0 u+ P
最小整解（±2,±1）                              最小整解（±7,±1）
+ H2 K" y6 s. C' C3 P, a: D) h* |                                                             ±7 MOD2=1" T; L9 u: G- f
: Z0 x% a% H- s5 `' N
两个基本单位：

11.JPG (3.19 KB, 下载次数: 313)

11.JPG

作者: lilianjie1    时间: 2012-1-4 18:53
lilianjie 发表于 2012-1-4 18:31
1 k2 h* S$ @  R' G6 t基本单位fundamentalunit ：3 L8 D6 c. P4 b2 T  r9 y) _' D
5 mod4 =1                              50 mod 4=2

) l5 h5 s/ r& x基本单位fundamentalunit

3.JPG (105.07 KB, 下载次数: 309)

3.JPG

2.JPG (140.29 KB, 下载次数: 312)

2.JPG

1.JPG (193.2 KB, 下载次数: 315)

1.JPG

作者: lilianjie1    时间: 2012-1-4 19:07
本帖最后由 lilianjie1 于 2012-1-4 19:16 编辑
- `7 ^# Z- F& A; v, S) V0 F2 z$ e& m/ c( P
判别式计算Discriminant* r5 ]$ f: ]& i4 e6 {; B& Y/ p' g
/ t- t8 _& d3 p' p& v8 P; S% Z2 |
5MOD 4=1
: J+ k$ ^! `. W/ ]9 G. B5 B: a5 A8 {& J6 [
（1+1）/2=1          （1-1）/2=0, E; }+ ]2 I0 L' ?
! A% P7 F6 t, L( [& j6 ^4 G
D=5) j0 D, `! m# J; `0 ]5 ~0 A% G
, U8 Y$ z2 w$ @" N' `! Z
$ S8 M# z" s& |2 m" P
50MOD 4=2. w/ G/ ]7 F  `& Q# p6 ?/ h
D=2*4=8

33.JPG (165.31 KB, 下载次数: 300)

33.JPG

22.JPG (137.12 KB, 下载次数: 294)

22.JPG

11.JPG (163.36 KB, 下载次数: 336)

11.JPG

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 8 d; @! ]5 b4 L5 s( u- T
$ }5 C2 |  S# d4 w* A
分圆多项式总是原多项式因子：
$ g4 r5 y& r- C. `C:=CyclotomicField(5);C;3 C7 P( v0 A% r* ~) X0 d
CyclotomicPolynomial(5);
' H& ~! G7 ~! j$ D

2 @" p% f2 a# y6 h  x. m分圆域：
$ q: u5 R2 K! x. @分圆域：123
% Y. |3 y" U- u) z) T+ V8 @" t. l# l6 B( }2 A) Z2 ]
R.<x> = Q[]
8 K' z; I3 N  j! a" }: rF8 = factor(x^8 - 1)3 ?$ z; j+ [5 r5 X9 Y
F8. L8 d  n) m2 `7 C( s

: o- @) @) I! Y$ }(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1) 6 \$ S' N/ q: Z4 o: X7 w4 B. J( f# `
3 _# L2 f, k; X) [( o+ t
Q<x> := QuadraticField(8);Q;
4 W; i' ]  G9 P1 Y' x$ e. uC:=CyclotomicField(8);C;
0 R1 M5 p; c1 C, u# F% xFF:=CyclotomicPolynomial(8);FF;3 A* F$ ^+ K' }4 k/ I3 W: ]5 i
# A1 v+ w5 D6 d1 s8 Y) C
F := QuadraticField(8);0 k) I) h* W* r/ K! Y6 Q
F;
; W! d. l) W* b- w1 DD:=Factorization(FF) ;D;3 t' Z6 n. ^. T2 p# L
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field3 h7 n  U3 [3 c. G/ u. d4 A
Cyclotomic Field of order 8 and degree 46 V% ]5 V  _# z/ q- ]) ~0 [  c$ j
$.1^4 + 12 b5 q6 E/ e+ b5 V  y7 {7 @+ _
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field5 ]0 _7 G: I3 m0 o
[9 R$ K$ S; X9 G
    <$.1^4 + 1, 1>
& }/ F& R+ C- j& []
0 n1 F7 y. w" \. ~0 v) X
: K0 A2 O' k; lR.<x> = QQ[]
+ ~$ E2 \1 b  \7 {F6 = factor(x^6 - 1)
; K$ n7 |/ G/ ]( J7 j7 H! V) X$ xF61 h1 I( g& T9 c" b% I& O5 G8 {+ _
- e, N7 j0 w# B2 n! ^
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1) $ j+ G) O+ ^6 K" t; b' d$ c  h
. \: l/ F" M! s# |+ V9 C
Q<x> := QuadraticField(6);Q;
" m! v6 A% w! n  p3 p6 \/ lC:=CyclotomicField(6);C;3 I1 F( b& x& k% s/ f4 n% J
FF:=CyclotomicPolynomial(6);FF;1 W0 V6 F. i2 J

9 g! t2 Z1 r: Y" l' YF := QuadraticField(6);
* J" e" T/ |; T# d  N, ?F;: d+ t' L" p/ B% i
D:=Factorization(FF) ;D;
6 A9 I, S# D) D( b& hQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
) N! G* t6 t* v: y( u3 N" `Cyclotomic Field of order 6 and degree 2
" X7 l* ~' h+ z0 x: u  W$.1^2 - $.1 + 1
+ t- o. G* k9 }, |( B" g) I, D8 [# mQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
4 z$ B) `3 E- z$ i8 w1 p0 A8 S[
" q( ?- G- |2 b# e& m    <$.1^2 - $.1 + 1, 1>7 S2 ^/ i) Z) b; X
]1 _) L  E: x  h

0 i7 E# }1 ]8 P3 Y3 YR.<x> = QQ[]) E. k* J& j7 w9 P& [5 X
F5 = factor(x^10 - 1)' ^: N. C+ |& }8 p2 Y* y
F5
6 I) S& `4 f9 Z( q' H, X9 R(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
( _' P+ t7 Q% O1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)9 Z" X( b0 ?" }$ U/ J) p8 P4 Q
, Q3 J2 I1 n* L
Q<x> := QuadraticField(10);Q;
  x: s+ z( ?0 d  s% _C:=CyclotomicField(10);C;
! c( S3 u' e, O4 D$ {FF:=CyclotomicPolynomial(10);FF;% M1 U, }# R: W3 P

& t2 @. Z$ e; G" uF := QuadraticField(10);3 s' x* V% v5 B9 ~6 N( Q+ Q2 @1 x) G- J
F;4 u! c; E0 J- \. m
D:=Factorization(FF) ;D;
4 T/ p8 Z$ W8 B+ _8 ~7 r; m. nQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field, s1 d, j, {& P
Cyclotomic Field of order 10 and degree 4, W' b, U& g: w9 ^  E, {7 O2 i8 M: k
$.1^4 - $.1^3 + $.1^2 - $.1 + 1
6 E: }; P, l  D, C0 xQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
/ q  V4 \! n* f9 h- K/ J[
% F& ~2 q5 D5 Q/ a    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1># ?& }+ _* i7 M1 Y* ~4 V8 T2 G& F% l
]



欢迎光临 数学建模社区-数学中国 (http://www.madio.net/) Powered by Discuz! X2.5