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标题: 实二次域(5/50)例2 [打印本页]

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作者: lilianjie    时间: 2012-1-4 14:05
标题: 实二次域(5/50)例2

9 Z% \4 [7 E; t$ I8 j! L9 {! L- V4 S0 ?
6 g& m5 \9 D. Z3 n8 h  s3 YQ5:=QuadraticField(5) ;
; U# Z) p9 a. l; _* N' xQ5;
Q<w> :=PolynomialRing(Q5);Q;

EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
+ }: U" A$ P  |  @' wM;
NumberField(M);
+ r% m4 L6 G- x$ E1 CS1:=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;
IsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);
2 g9 l5 C! A% n7 t4 RFactorization(w^2-3);
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
  I& ?( r; {' `$ ?0 {3 N& PFundamentalUnit(M);
- l# P9 ]" L9 C, Y( _4 |! J  wConductor(Q5) ;
9 K# B$ i! E& v$ w8 v' f: f$ g* oName(Q5, 1);
Name(M, 1);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
9 _% H$ A2 \5 z. G
PicardGroup(M) ;
PicardNumber(M) ;

! G8 B* h, m" k6 K3 B0 O
9 `: l. ]$ g* q. DQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
$ y2 o) d$ y5 o4 J# D: I

5 F2 w) x- ^6 _NormEquation(Q5, 5) ;
NormEquation(M, 5) ;

& J: D! c9 W( G3 J! i6 a. }3 A
6 B4 T- s& x# q2 M+ C* vQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
+ p1 d( Z; t% jEquation Order of conductor 2 in Q5
Maximal Order of Q5
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
$ ~: W1 V5 h9 z3 QOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
4 q: \' W! S0 {( U& F2 V& W( L+ ptrue Maximal Order of Q5
true Order of conductor 16 in Q5
: G# M% y1 q4 z7 X$ Y, H/ r7 otrue Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
; i1 M. w8 ^5 H; W* J. T( b% {[
    <w^2 - 3, 1>
]
5
8 k* w1 l  x- m) }/ m9 B5 z1/2*(-Q5.1 + 1)
5 L6 b, H0 T) o-$.2 + 1
5
Q5.1
  Z+ O4 b. v1 U$.2
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
6 a% K* y/ `5 x! q& y7 c4 b' q/ ]/ _Mapping from: Abelian Group of order 1 to Set of ideals of M
1
; A( a/ f# y: d8 @8 E1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
$ O# X% h7 [9 W1 z! T1
, j; T: _$ @! G4 h# Z1 y8 e; uAbelian Group of order 1
& v1 G" N% _8 Q/ K5 N) {, Z! A# ~Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
Abelian Group of order 1
/ _/ k6 h$ F% L  P# }8 oMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
5 j4 P, g! U" s; ltrue [ 1/2*(Q5.1 + 5) ]
true [ -2*$.2 + 1 ]

- `- l0 a; Z- M3 b6 r


+ x3 V1 \7 i) V% D% h( b/ M
+ }4 m' L* {' Z( ]
* _" C! |0 V, h! Q+ p, u' m2 G& ?

# C8 u+ y; H! L& k* d( e


4 o+ u) }  z" Q1 C7 ~) D==============

; O' w1 H3 Z9 ?- N6 r7 [Q5:=QuadraticField(50) ;
4 x+ n3 t; {  ^1 ?9 X# h" EQ5;

Q<w> :=PolynomialRing(Q5);Q;
; p7 b! ^2 d: V. gEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
0 k3 r5 k2 ~. e- U# JM;
! @( u4 l2 u4 ?+ s: `: t! rNumberField(M);
+ e6 B2 [4 d- i& ^+ D' S4 nS1:=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;
IsQuadratic(Q5);
2 v+ J6 q0 L- fIsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
- F8 i2 W; @" ^' w* V  A6 }IsQuadratic(S625888888);
Factorization(w^2-50);  
Discriminant(Q5) ;
! l$ Z. y8 E4 r" L+ H+ @FundamentalUnit(Q5) ;
& L5 s+ u: t. aFundamentalUnit(M);
Conductor(Q5) ;
- Z' p( P4 I/ T' A' F
Name(M, 50);
0 U  c4 v: {; K+ h; H0 W/ H/ \/ V* XConductor(M);
% ^" k1 Z/ n8 d/ {: kClassGroup(Q5) ;
7 Y) M% F$ w' b' M: \6 @3 m2 RClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
% }; z/ H7 X- I9 v' qPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, 50) ;
NormEquation(M, 50) ;

" Z1 S# {; W) P0 @Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
5 p% a1 I  {) TEquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
# D: Q1 R9 K0 \" _* \  TQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
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
+ P, ]5 ^0 i1 J9 R; _) J[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
8
Q5.1 + 1
$.2 + 1
8

, ~! B* t1 m4 S>> Name(M, 50);
       ^
# S4 C- }, q3 @2 b- I6 C0 V. \Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
& n/ \0 K+ D) }+ u1 Q3 r! w+ m
1
$ V4 f. o, P6 i/ n1 x& rAbelian Group of order 1
4 {4 t; ~5 z) d" j# b2 n5 SMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
, v# o' z0 r2 a( J1
1
. e; w2 ^0 d& p+ ZAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
! d* P" ]( X  `) l1
  e6 C0 |0 f3 c6 c; ?Abelian Group of order 1
: P" t3 l& \- d) d2 S' ^# e9 i2 i( RMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
) N  _/ E: P% SAbelian Group of order 1
7 \( v' t7 u  Q0 o& J2 JMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
/ K7 J/ R* W! ?, ~: S! m3 r1 ?8 given by a rule [no inverse]
true [ 5*Q5.1 + 10 ]
2 w" _; A9 T( \0 I4 d) gtrue [ -5*$.2 ]

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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

+ u6 Q0 d. A, r; Q+ ^1 M7 |$ D
基本单位计算fundamentalunit ：
5 mod4 =1                                              50 mod 4=2
, D! K, \% ~& w( o, X* t+ T
3 Q; A, p: V! n1 n# ^ x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.
- `+ h  [0 ?0 ~! r+ i6 ]
) }6 c# d# S# m: G% t- u* P; J
最小整解（±2,±1）                              最小整解（±7,±1）
4 d1 N+ a! E- r" V$ n- Y                                                             ±7 MOD2=1

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
基本单位fundamentalunit ：
& Q. d& X1 e/ N# M  g6 {5 mod4 =1                              50 mod 4=2

( c4 _8 t: A9 |" Y5 E; f# a( B基本单位fundamentalunit

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作者: lilianjie1    时间: 2012-1-4 19:07

# f6 R0 C5 {- m
9 K% p1 P* |4 W判别式计算Discriminant
' y' I# [. |* `% H& S1 R( U/ G
0 [/ S: @- E. V3 F5MOD 4=1

（1+1）/2=1          （1-1）/2=0
0 y- L) l! W8 ~2 I
D=5


" Y* P/ c. x/ C2 W5 X1 Y# c( W0 p50MOD 4=2
' A. }: n, q/ V3 W& LD=2*4=8

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作者: 孤寂冷逍遥    时间: 2012-1-5 08:37

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作者: lilianjie    时间: 2012-1-10 17:49

lilianjie 发表于 2012-1-9 20:44
( c8 z" ^* _2 F5 ^4 |- p
. E5 d  a+ m7 L( J分圆多项式总是原多项式因子：
% c' x' ^8 [3 ]$ W8 \  X5 [C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

8 l" E* E" Y! k分圆域：
) X1 B; j7 L4 d' b, v分圆域：123

% b# Q' J. ^) z' n1 P  P0 a& W6 zR.<x> = Q[]
7 t! r; R0 [# {) _" VF8 = factor(x^8 - 1)
, [2 H+ U6 c4 `+ P5 A7 fF8
2 Q; s# q0 }4 y
# v" y5 ^, y' t( j) j/ O: V6 Q(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

Q<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
* P8 @# R+ S3 P8 M7 E( Z% QFF:=CyclotomicPolynomial(8);FF;

F := QuadraticField(8);
8 n; r5 F  `* g! ~7 z0 VF;
+ q* Q# d$ Z) u! ?& D- QD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
& S7 a. V4 P+ D6 @9 m  vCyclotomic Field of order 8 and degree 4
) s) R- s+ Q7 o7 F$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
8 O: q$ ~5 ]3 D8 x, e2 H+ {5 \    <$.1^4 + 1, 1>
]

5 d% Z( i. t1 Z+ J. s# _, \R.<x> = QQ[]
F6 = factor(x^6 - 1)
F6
! n1 `: b9 @- j
! x, Y- o& ~* q& A5 M(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
6 A" b) k6 s0 U1 g4 {  k/ X, A
Q<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
3 N$ a* c6 D0 CFF:=CyclotomicPolynomial(6);FF;
2 T$ J! I5 Y' D8 i' L
F := QuadraticField(6);
F;
% J. e+ E6 n# u( w1 cD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
9 i* K0 P! A# K4 W2 zCyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
% K  E! ~: Z/ a* b% j: |[
    <$.1^2 - $.1 + 1, 1>
3 b3 s! q' F& q. z]

% y3 b; o+ m2 o) L8 kR.<x> = QQ[]
F5 = factor(x^10 - 1)
, t  P; x# d$ O& rF5
(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)

$ z4 g  k; y% ?8 h  [# X1 ?Q<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
! {8 }0 `7 y6 u$ B; `FF:=CyclotomicPolynomial(10);FF;

# h2 a/ r2 `* c/ i' |8 O3 j* TF := QuadraticField(10);
- N: Y4 w6 g. K: P, ]F;
D:=Factorization(FF) ;D;
+ o9 T/ v, q0 B4 r6 vQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
2 F2 I6 ]: ^, t( y: K' G5 c/ bCyclotomic Field of order 10 and degree 4
' F, Z& }0 [. N$.1^4 - $.1^3 + $.1^2 - $.1 + 1
3 A8 W: S1 ]+ a' K7 a* bQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
% y; K* ?. j$ x7 P( ?    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

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