实二次域(5/50)例2

lilianjie

  \* O. N: [0 ], c, {1 N' O
- `2 v7 y0 A* j1 gQ5:=QuadraticField(5) ;
- s: V0 v) f% f6 W6 HQ5;
/ Y% i. t& R: a3 e6 c1 BQ<w> :=PolynomialRing(Q5);Q;

EquationOrder(Q5);
- Y( G  J/ C3 g- ?! p; c" KM:=MaximalOrder(Q5) ;
6 N6 E, r, ~4 b; GM;
NumberField(M);
, g& A8 F3 H0 n8 s+ s1 }' aS1:=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);
Factorization(w^2-3);
" f% P+ H0 C8 B$ S. fDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
Name(Q5, 1);
; v! _7 R6 ~3 M. w/ l, qName(M, 1);
$ |% K+ ]/ X0 W1 q1 P( v2 dConductor(M);
ClassGroup(Q5) ;
* N' R% x1 S! ]' sClassGroup(M);
, x) f6 o  h0 I1 g3 jClassNumber(Q5) ;
2 ~9 Y8 J# ~, M4 q; x. @ClassNumber(M) ;

( N; D# b/ K' S8 }+ {/ T* C, ePicardGroup(M) ;
PicardNumber(M) ;

( h; f4 K$ i2 ]' P! Y
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);


NormEquation(Q5, 5) ;
NormEquation(M, 5) ;
! `6 }, z; I% u, N& |$ z$ Z
8 c# k/ S7 Z5 f3 M6 R
& O* ~7 F3 S: g* G5 h6 I+ VQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
+ N& W4 G4 G9 q; T. E: w. VMaximal Order of Q5
1 h' ?; d4 W5 p# p/ m, ?, d1 VQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
& g7 F% u6 ^: e- ~2 ?Order of conductor 625888888 in Q5
9 n" _) y9 f: h" n7 utrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
" P! [' u/ [- H" j9 }true Maximal Order of Q5
& h* Z1 ^/ \! O$ `# H. `0 ~# J6 jtrue Order of conductor 16 in Q5
4 p$ V7 g0 R& |* P* dtrue Order of conductor 625 in Q5
0 P, b  O0 o7 n5 S  t; {/ }- ktrue Order of conductor 391736900121876544 in Q5
1 S- y% C' ^. R9 L" H  L1 j; K[
    <w^2 - 3, 1>
8 H+ W, h# `+ ]1 Y" T]
& d" L; y2 K) d: U1 Q) }( O5
1/2*(-Q5.1 + 1)
7 J; m" u+ z  \: ?6 {4 C-$.2 + 1
5
Q5.1
$.2
# d4 f8 y) o. @* [1
; v1 b5 N. L' R  aAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
" l, }$ b$ s0 q7 EAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
1
Abelian Group of order 1
# ^' O/ @7 j) u* L/ ?Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
' Q5 |/ N7 ]" \& ?2 E$ AAbelian Group of order 1
% g1 I# n  D2 E' O. g' Z+ f" {Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
0 V; d3 W% t$ |/ ?% I5 X/ AAbelian Group of order 1
& {+ T. P% ~5 ^; K/ ~+ C4 V# kMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
true [ 1/2*(Q5.1 + 5) ]
# p" F* y# W; m5 l: g) Utrue [ -2*$.2 + 1 ]
. C. ^3 q  Y; j5 n4 p7 N6 D" ^
( y7 _# U2 ~3 v5 p) w7 a/ k
8 P9 N0 B/ v9 s1 V  j
& _& Z7 M) n# S8 ~
  K8 k1 ~5 R" t( B% {% [1 n" i

3 u6 Q3 a4 s8 E* q4 N  _

5 X2 z" D5 ^, V# b* K
9 F- N2 C3 X, F' S/ n) C/ b

==============
( [5 u2 q, j, T. I5 O1 B
Q5:=QuadraticField(50) ;
; f+ P3 s8 G9 i+ v' F2 xQ5;
9 M" Y; C; `0 k) G5 [
Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
# B5 E% X8 y. l1 F& S6 ^4 kM:=MaximalOrder(Q5) ;
9 {7 _5 r  A3 |2 DM;
# A! F2 E# m" m1 O  a$ o" V3 |1 qNumberField(M);
0 }3 y. F/ C* W! LS1:=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;
6 ~4 w: n5 X6 i9 }( e- e/ KIsQuadratic(Q5);
( {' C' a3 Y7 F. xIsQuadratic(S1);
IsQuadratic(S4);
1 R! N  M9 g! z: m% e8 W" iIsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2-50);  
" D, D2 [  h, B8 o$ j! vDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
/ @4 j  H& w& qFundamentalUnit(M);
Conductor(Q5) ;

8 l6 V  {4 m9 c2 z+ q3 |Name(M, 50);
Conductor(M);
8 _7 ~; M9 W5 @: LClassGroup(Q5) ;
* M2 @  N: P# b" p& QClassGroup(M);
; Q7 h; H' N2 C  Y$ E2 z& W# iClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
& V/ V) h- G* {PicardNumber(M) ;
! P, _# E/ q" v$ }& e5 {, ^6 W
QuadraticClassGroupTwoPart(Q5);
2 v; [7 _  s! p* p- |& O0 i% sQuadraticClassGroupTwoPart(M);
. x! b8 R& n& PNormEquation(Q5, 50) ;
! L2 G- W5 s% a% n0 \7 Z7 n5 gNormEquation(M, 50) ;
$ u8 ?% P$ ?8 l6 f1 D! E, n
" R$ F8 ]- E6 I* p& y/ ^+ ]Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
8 s& @" A. L7 o, YEquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic 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
! C, c) h  o6 n/ u" Y3 L) @true Order of conductor 1 in Q5
2 s9 Z0 Y8 y# E7 H8 I7 ntrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
) Q6 U7 x* X+ K8 A" I4 o0 a]
8
Q5.1 + 1
$.2 + 1
8
( E' b; ~7 q! D" z+ [
>> Name(M, 50);
, L2 d& o# _4 @       ^
Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
+ ^9 z1 j. e1 T' d* J9 o
1
& U+ t- A* X, [) w7 rAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
4 t9 |8 U2 H. r- q8 I8 i( wMapping from: Abelian Group of order 1 to Set of ideals of M
1
0 a9 |( t2 o2 \" s, w1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
: o8 z5 s8 A& I$ O5 l- B$ `& g& y: {inverse]
! e" I& G$ Z% _! V1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
/ G, m0 |5 `1 s2 t/ M8 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
. v8 F9 Z9 m8 n4 N5 X8 given by a rule [no inverse]
true [ 5*Q5.1 + 10 ]
true [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

0 C, K$ ~! K3 V. d基本单位计算fundamentalunit ：
# A# o7 r. N- L3 c" `( _9 ]! F5 mod4 =1                                              50 mod 4=2

x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
, E/ j" _( j: U) I" r/ M/ R3 G' K  U1 i x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.
0 ?0 x; b9 s9 P$ W- g; X% n

最小整解（±2,±1）                              最小整解（±7,±1）
                                                             ±7 MOD2=1
( E& O7 K5 ^) d4 Q4 g8 Y( U
两个基本单位：

lilianjie1

lilianjie 发表于 2012-1-4 18:31
基本单位fundamentalunit ：
4 s( k. w7 ]. h& h5 mod4 =1                              50 mod 4=2

基本单位fundamentalunit

lilianjie1

. C7 G1 |" K. g! [; B( M9 S( h* k判别式计算Discriminant

4 {9 P2 S# v) t/ S5MOD 4=1
6 ~; [/ T1 p4 w
) g8 ]6 U: R$ D（1+1）/2=1          （1-1）/2=0
& D7 e- g& }) ^% V
D=5


50MOD 4=2
/ S+ P1 B0 O5 ~/ E7 {, [D=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44
. s: O. V4 C  A! T
2 P+ q$ e. f* o' }" z% _分圆多项式总是原多项式因子：
C:=CyclotomicField(5);C;
/ A$ i: D, j7 @% a5 b! mCyclotomicPolynomial(5);

6 g& @. b" N6 B; k. l; T4 z分圆域：
  t* Z( p3 X: `# O+ m6 E0 H分圆域：123

R.<x> = Q[]
/ l) V* U% y! B4 |: ?$ A0 ^F8 = factor(x^8 - 1)
5 |$ p3 c0 w$ Q! O1 h+ L6 _F8

(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
4 X8 p1 t& P1 f6 m& E- u- J
Q<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;

F := QuadraticField(8);
F;
6 t( D8 ?+ i  S5 ~D:=Factorization(FF) ;D;
/ x" _0 ~  P7 B7 B6 i" I3 ]4 QQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
$ D: |$ b- `  M- F& {# C+ K7 YCyclotomic Field of order 8 and degree 4
$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
    <$.1^4 + 1, 1>
]

R.<x> = QQ[]
F6 = factor(x^6 - 1)
. T: b9 i% ?5 S0 z+ yF6
7 ]+ G( x  f: E
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
  p7 }) B. Q: u. `. ]
6 E2 t! s) g( j! Y1 P# N: x) qQ<x> := QuadraticField(6);Q;
% n! `2 U7 O9 G: DC:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
4 z" F: Y' {) |& f" r" H* X; aF;
$ u" f* J& N! W& }" h0 H7 ~) sD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
+ ]+ G( _& p6 p1 BCyclotomic Field of order 6 and degree 2
( y5 Q3 C' P$ ^: a/ h5 ?+ U7 }$.1^2 - $.1 + 1
) l6 x# C# R) U6 }0 F. n: j! K4 QQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
  J, x1 B3 o# J    <$.1^2 - $.1 + 1, 1>
]
6 @2 x  Q, g+ [) |
& L/ L% ~1 ~4 o" U0 DR.<x> = QQ[]
" m  A# c9 v8 ZF5 = factor(x^10 - 1)
F5
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
9 _  l. ?$ V  `& @  K1 I: M0 a
4 a7 l& Q8 Y% Z; L" }Q<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;
7 z- h/ l) u1 R$ L" Z4 `
F := QuadraticField(10);
" H1 E3 p9 u9 N7 `F;
& I0 V; B6 R* Y& i2 GD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
$.1^4 - $.1^3 + $.1^2 - $.1 + 1
; K3 V( r! b% R8 mQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
1 E: A- S; \" H[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]