实二次域(5/50)例2

lilianjie

Q5:=QuadraticField(5) ;
$ D. B6 Y6 y; g/ oQ5;
Q<w> :=PolynomialRing(Q5);Q;
0 l( P. Y" D0 S
7 }# {3 M3 O& {# C" m# e: p( sEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
, k: v" R, l# l, D" i; w/ |; W9 Y9 pM;
* ]2 f5 e: y) f* r# O9 r$ SNumberField(M);
8 j+ k, h5 `- h% bS1:=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);
) Q1 [+ t9 o' M7 MFactorization(w^2-3);
+ n" \; I. T+ N0 V, S+ qDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
( \2 q" B* O, a  }) ~FundamentalUnit(M);
Conductor(Q5) ;
Name(Q5, 1);
Name(M, 1);
! h2 i5 I: }1 O- h: {8 Q0 XConductor(M);
ClassGroup(Q5) ;
( ^7 K6 ~5 e) E" _5 ^7 {/ QClassGroup(M);
& c1 [) x: g; }, A) O$ TClassNumber(Q5) ;
ClassNumber(M) ;
3 S/ U8 m2 N4 n+ y& T( b0 X
PicardGroup(M) ;
3 j) ]7 q: z* CPicardNumber(M) ;

4 ~7 Y* v: P; N; V0 ]
QuadraticClassGroupTwoPart(Q5);
& ~) @, h/ J, f1 W6 v9 y4 RQuadraticClassGroupTwoPart(M);

% K/ e7 J) ]6 O+ c+ \7 x$ B/ h
NormEquation(Q5, 5) ;
NormEquation(M, 5) ;

: O! c& f' o8 A/ h' E
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
3 {( I& a1 H' V6 BUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
1 o- N( l2 n: M; aMaximal Order of Q5
0 f2 Y' j" y) Z5 d1 {' w7 k, BQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
' @& S! J% J+ i' M! A5 l1 x( FOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
8 y9 Y% K9 X% v" \( k  e! ^& Y' jtrue Maximal Order of Q5
true Order of conductor 16 in Q5
true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
[
* C5 T) N" S8 ~1 w    <w^2 - 3, 1>
]
  C. a. f, x" m6 s* I0 f/ q6 h5
, B$ g# Y6 ?3 T5 A: Q1/2*(-Q5.1 + 1)
5 g0 k% \4 r% W- ?-$.2 + 1
% c* |8 s3 R6 J$ N% ^5
+ a1 H2 i% o5 m; bQ5.1
* h1 X/ s2 x! |$.2
8 g* C/ }' v$ h8 v1
3 M' A4 h+ G! @( x9 gAbelian Group of order 1
Mapping 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
8 e: P. H3 l* C+ r: E9 {! ^1
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
7 y, I9 A: c- G1 L& I# c2 Z1
1 u  S# {, w& W5 T4 ^6 ~0 c# uAbelian Group of order 1
. Y  }" s$ p) K' W8 MMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 I2 _9 r8 e, K0 m3 x5 given by a rule [no inverse]
; Q# g1 B0 M7 u5 CAbelian Group of order 1
' I6 L- v' t1 d6 WMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
" o  O( @! T4 I4 E' g5 given by a rule [no inverse]
true [ 1/2*(Q5.1 + 5) ]
true [ -2*$.2 + 1 ]
7 }9 x7 s5 K6 J! r





( B8 f7 `- Y' K7 Y8 Y+ F6 Q
6 g( e' e, C& X0 Q6 ~1 K



4 }$ J7 h' J. [2 b/ G- x; J, F/ @==============
( [5 G5 T; z/ Y+ Q
5 Q3 N5 s. {- ?# b1 uQ5:=QuadraticField(50) ;
Q5;
& f8 x& F5 A) d: l7 @: R* k
3 _7 i4 X: ^0 a! h" W& sQ<w> :=PolynomialRing(Q5);Q;
& u9 h& U) A  B% n  f& v& J# MEquationOrder(Q5);
) |, Z6 a+ Z' j! g  }; n+ M. E6 UM:=MaximalOrder(Q5) ;
( }9 u  ~5 d, _/ W9 S  qM;
, Z; @. Z: n1 `' H: |5 sNumberField(M);
; w5 i4 q! k2 H- jS1:=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);
% V, {# F7 _% O/ {0 x- R2 f2 ?IsQuadratic(S625888888);
Factorization(w^2-50);  
Discriminant(Q5) ;
! S  M: w5 `5 B5 V3 @FundamentalUnit(Q5) ;
FundamentalUnit(M);
, @' U' P4 B6 z* bConductor(Q5) ;

' ^2 ~  f* C9 Z% NName(M, 50);
Conductor(M);
8 H4 ^4 V! L! u3 P( x7 ?! YClassGroup(Q5) ;
8 q6 M+ m7 M9 X. p2 yClassGroup(M);
ClassNumber(Q5) ;
# X$ X4 T  Q+ @/ j1 Z' JClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
1 i3 c: O/ _, x2 M8 VQuadraticClassGroupTwoPart(M);
$ O7 p% [0 e, h0 E! P" }1 KNormEquation(Q5, 50) ;
NormEquation(M, 50) ;
* \+ K" h: Q& ]
! w# Y0 w) @: K7 `Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
  a; \1 g6 K" X; \Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
  k! R+ g9 O7 M1 t9 w. HMaximal Equation Order of Q5
1 L( I2 A  j. A. @4 JQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
' U2 g0 U1 {6 `% a8 WOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
9 P6 ]% R- n1 W4 j1 J# m$ rtrue 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
[
    <w - 5*Q5.1, 1>,
. ?0 l& d) L. X0 {# O' F    <w + 5*Q5.1, 1>
]
! a, T% g) f* X% E3 {1 R8
  _: M: @: J9 b* i4 S, E; CQ5.1 + 1
1 _( d& `/ B. c9 H' Q' {( s2 Z$.2 + 1
8

: X2 C" [0 Y1 D4 y: c, s>> Name(M, 50);
       ^
Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]

: g! ]  \) s4 T' Z' @1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
  D) i- v+ T6 `8 V7 R6 e- uAbelian Group of order 1
. f! h5 `; l6 r  Z6 @; Y/ l5 eMapping from: Abelian Group of order 1 to Set of ideals of M
6 P3 r/ z5 c1 z* N& r+ L1
1
/ [6 V0 Z1 |* e9 {Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
* a+ H" a& D" d3 x  S: z1
Abelian Group of order 1
5 ^8 ], T2 j# s( _3 k0 t4 V6 WMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
* E6 W8 S7 Z2 ?- [$ _9 c7 j; {Abelian Group of order 1
. e/ F! h3 u1 i- YMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
1 U$ u* F* n  \4 S0 H# N8 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

( x3 s/ z$ f- L
基本单位计算fundamentalunit ：
6 G' S5 D1 m, E/ e& P' d" f5 mod4 =1                                              50 mod 4=2

$ A) M& g7 b2 i x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.


最小整解（±2,±1）                              最小整解（±7,±1）
1 E/ Q& f' k% `7 `3 Q. N                                                             ±7 MOD2=1
+ m- w4 i+ v1 p! \  c1 A( l
$ o1 b# s" H2 _' O8 ^) z; V两个基本单位：

lilianjie1

lilianjie 发表于 2012-1-4 18:31
& Q$ e% L1 W8 S0 w0 @5 H基本单位fundamentalunit ：
5 mod4 =1                              50 mod 4=2

基本单位fundamentalunit

lilianjie1

; o8 {+ |3 C, ]判别式计算Discriminant

5MOD 4=1
5 [+ u/ x6 b6 k6 [% Q. U
（1+1）/2=1          （1-1）/2=0
$ e5 [  ?# }- a
3 M- L  t( w$ A2 E: @; J: ED=5


, M0 L) M# |8 R. M50MOD 4=2
! ?" a) C! s" k6 A" M3 s% FD=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44
4 d8 e( {& ^9 ~" A4 S9 A
" _( E+ k: w' Y' o' i: c& |分圆多项式总是原多项式因子：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

分圆域：
分圆域：123
  x  H( W5 s+ R) F3 b, H: J. G
" f4 m" e! x* R7 S' y9 tR.<x> = Q[]
& e& B5 r/ h# x: [8 ^F8 = factor(x^8 - 1)
$ D$ V* [0 Q$ b+ G, z$ @0 E. mF8
" N3 _: l' B. H% u, o
(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  D  X& ]% y. m* nC:=CyclotomicField(8);C;
( A7 V$ S& J  _8 rFF:=CyclotomicPolynomial(8);FF;
4 s( X% m% Z( P7 @! w& a
- c9 h" n7 B9 CF := QuadraticField(8);
F;
% V* {2 p1 I. r! w7 |& a( |D:=Factorization(FF) ;D;
) M0 L' n( I/ h& |; t, J2 ~% zQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
% K9 v* [# m4 v  c5 c  A0 ~+ f$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
! F) q- W; P9 \3 Y! r$ j    <$.1^4 + 1, 1>
. v  Z* z. b- T, J& N/ m]
; i( A0 z  J3 h7 p0 |
/ c. u+ `0 {- T* aR.<x> = QQ[]
( Y/ W+ {. J, PF6 = factor(x^6 - 1)
, I* t: G1 n! a. B0 O  p& U1 HF6

(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
: w: ]$ U# A) g! V) _6 g
Q<x> := QuadraticField(6);Q;
2 M9 E: u; U* t! aC:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
F;
$ Z4 l' e" p2 g1 b, u# I" @( `D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
8 d9 F* B: N& f% b* F$ n2 ZCyclotomic Field of order 6 and degree 2
  h. c3 z  S5 Q! l" _, v: p$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
6 j5 o. Q! H1 d3 ]; o2 Z$ L6 L    <$.1^2 - $.1 + 1, 1>
7 {. J! {, k3 x( s! P]
- _6 m9 ]( R7 E$ F- v& H
R.<x> = QQ[]
! O) v- _0 R. p$ xF5 = factor(x^10 - 1)
% d$ t# P  S5 UF5
! i$ S) y# O9 H(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
' E( ]- g1 t# }8 M" c' T1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
$ T7 W# [  P) B9 h0 q4 B  I9 `+ @, y) |" a
Q<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
& W* Y% @5 Q5 O1 Q  _" JFF:=CyclotomicPolynomial(10);FF;
9 r; e* _0 q% c4 ~3 a: Y/ ?
0 a1 e2 Q' C- N' T* [F := QuadraticField(10);
9 L; K6 j) H  X  L* d2 MF;
2 T" @8 i* T5 U9 UD:=Factorization(FF) ;D;
1 ~' ~8 J( N$ s0 P7 mQuadratic 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
, r* ?4 \# u. P9 fQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]