实二次域(5/50)例2

lilianjie

" H3 J3 {( N  M/ u
0 ~4 x4 k" M; W! N7 pQ5:=QuadraticField(5) ;
Q5;
; r2 t; d- H; C; PQ<w> :=PolynomialRing(Q5);Q;
6 s# x8 j! v& ?1 }* c" O$ j# A
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
# @% e. ]9 D' oM;
NumberField(M);
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! x2 t/ a% J4 d- TIsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);
- O: a# O; T8 U+ C) f( O% zFactorization(w^2-3);
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
0 l0 X3 F2 K, ZFundamentalUnit(M);
; k# b9 v* @; ?1 H# W% ~0 _  GConductor(Q5) ;
Name(Q5, 1);
8 C6 F5 J/ m9 J7 N) c1 `4 UName(M, 1);
Conductor(M);
ClassGroup(Q5) ;
1 n- \* O3 @% K- lClassGroup(M);
ClassNumber(Q5) ;
" H/ c( ^8 h1 i/ x1 XClassNumber(M) ;

" R& K- b; s$ }; _- GPicardGroup(M) ;
PicardNumber(M) ;
, C5 g$ o& E# Q

4 v- z: s; _8 }9 O; z- m! x# ?" jQuadraticClassGroupTwoPart(Q5);
6 c9 Q7 n8 ~# @" ?: k& xQuadraticClassGroupTwoPart(M);
9 |( J  e( y; n4 \" k
" {& N$ c4 ~6 |; ^8 Q$ v
3 x" f) i' M) D4 l4 ONormEquation(Q5, 5) ;
NormEquation(M, 5) ;
# a4 r# d6 x% s4 w: A) c
6 i$ B1 f$ z4 J' }0 g. a: U
) \) ]% Y" C' L) NQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
2 L# k* }* O) x- Q/ Q- VUnivariate Polynomial Ring in w over Q5
% v- g2 o0 j0 R& NEquation Order of conductor 2 in Q5
8 E" {& }& t2 C2 n% sMaximal Order of Q5
& I& l  K9 Z1 U3 v) xQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
true Order of conductor 16 in Q5
true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
[
1 E" _! K% p: {+ U  O. O- ~    <w^2 - 3, 1>
  ]& Z. h* E, V, p1 @% j8 i8 r) J]
5
1/2*(-Q5.1 + 1)
-$.2 + 1
5
Q5.1
$.2
) X  o. H7 r! K3 u1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
" Y- S1 z/ V  X) w, T# qMapping from: Abelian Group of order 1 to Set of ideals of M
1
5 E- r8 j' Z; Q) J' u1
0 Y, s" C, s/ A! {( GAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
0 f0 C: r3 y. a- NAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
9 `9 _1 V) E; s5 given by a rule [no inverse]
7 H! X- w, b: E" FAbelian Group of order 1
! y* Y- N7 Q9 J- r; H/ [Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
6 b# O3 ^, _! }; Ftrue [ 1/2*(Q5.1 + 5) ]
true [ -2*$.2 + 1 ]


9 O! W. G& L/ A' x9 R4 D( _
$ w; e, [6 `: B- j
# J0 y: d6 ~* T; J- L- x


8 f+ r0 `) n5 \
. G7 V: O3 z/ Z# P( ^1 Z0 {; C+ ]6 I


==============

Q5:=QuadraticField(50) ;
Q5;

6 t6 E+ H3 i$ n1 PQ<w> :=PolynomialRing(Q5);Q;
6 r, E4 Z+ M# I! b( ^/ AEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
, B+ L. I/ p1 C: L# [" ~M;
- z# M4 Z8 i1 c6 \NumberField(M);
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;
IsQuadratic(Q5);
IsQuadratic(S1);
1 z' Q/ h$ y+ d2 o) UIsQuadratic(S4);
IsQuadratic(S25);
3 c+ y2 k  i( z$ j, P- V, k, W( _IsQuadratic(S625888888);
  J( p* j9 N; d& s2 X# e5 g) U3 gFactorization(w^2-50);  
1 A+ C4 Q% e. g( \- \3 j. EDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

: a; P1 B3 B6 {0 o8 W1 R" G" t( SName(M, 50);
8 r! }3 r% q3 G9 v( wConductor(M);
9 h7 N# @- z4 |# f% t: x& E$ ?3 ^3 WClassGroup(Q5) ;
* S" R+ {8 l3 s8 _, ?& g* yClassGroup(M);
ClassNumber(Q5) ;
7 n3 s% @0 K% \. cClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
( @/ i' Z) a0 p8 `3 k
: w/ R; ]% b, a  vQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
; i. ^+ {+ @4 Q, m* E( x4 X9 l5 \& A( kNormEquation(Q5, 50) ;
% ^0 ?) [& G" Q9 R: RNormEquation(M, 50) ;
. ]* B  v! D' H  g" l/ n
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
1 |. A8 g$ V/ DUnivariate Polynomial Ring in w over Q5
Equation 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
7 X0 Q) u& c/ Strue Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
# z9 T6 v" g; B% u& i. Z$ C5 Atrue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
$ r: W. ~& }0 {6 {true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
, `% l1 t2 `) p6 q* \    <w - 5*Q5.1, 1>,
6 l9 G0 L, e. m: x) |6 B) Q    <w + 5*Q5.1, 1>
" C! X0 ]6 v# \]
" o4 s/ t9 m" w2 F9 S( n8
/ ]/ ?. |2 D, B; d/ Z2 `Q5.1 + 1
$.2 + 1
! s: H" K$ ^' m/ a8
6 G2 |8 E% w5 \* Y: G. B
>> Name(M, 50);
       ^
8 {* X1 p/ A% F3 Q; vRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
! p- k  [' r9 c4 G6 d
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 l/ g4 B* Y( w$ s4 Z0 LMapping from: Abelian Group of order 1 to Set of ideals of M
: k$ j- q) d) R* x# n+ U; _1
. n" q- O  f1 }) s6 P1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
& _/ s+ }) A( W% W& Q) |, ]8 given by a rule [no inverse]
true [ 5*Q5.1 + 10 ]
1 G3 E( f* q  }" [6 ltrue [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

基本单位计算fundamentalunit ：
9 {2 w0 \2 w3 r. b5 mod4 =1                                              50 mod 4=2

x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
" o5 R: k* o. `$ p. w x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.
+ H% L) e% ~6 m4 l3 z

最小整解（±2,±1）                              最小整解（±7,±1）
                                                             ±7 MOD2=1

两个基本单位：

lilianjie1

lilianjie 发表于 2012-1-4 18:31
基本单位fundamentalunit ：
$ _" d  C( k1 `4 ^% A3 X: \5 mod4 =1                              50 mod 4=2

, O1 b: i2 R6 f7 u$ `基本单位fundamentalunit

lilianjie1

6 g) W5 c# C& }+ j7 e判别式计算Discriminant
8 o# a7 V2 I5 G0 u, v& O! {1 k$ Z
# l& f! u: x7 \; a( e5MOD 4=1

# D7 Q" j  ?% @$ a' R8 h- M（1+1）/2=1          （1-1）/2=0

D=5
; H4 H& D) T' T4 l

50MOD 4=2
D=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44
/ a( B$ d1 ?8 G
分圆多项式总是原多项式因子：
  V' K/ Q) F& i, r" ZC:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

  g$ l+ Q7 g8 G( ]
. c0 c2 G# c+ Y; b. v% G3 [分圆域：
9 W6 E0 f/ j0 H' u  V9 O- m, F分圆域：123
4 _( U' C; z! ~6 e
R.<x> = Q[]
9 @3 e) Z& d  N' h' s  jF8 = factor(x^8 - 1)
F8
$ {0 C$ o9 f8 G& g0 ]/ H2 k
(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;
) H, E2 M# A6 r1 ~3 h" UFF:=CyclotomicPolynomial(8);FF;

# w& Q% A0 A) t6 tF := QuadraticField(8);
1 B7 e" {0 Z  E+ F/ \' UF;
, ]4 M  h8 I$ x  u- e' CD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
$.1^4 + 1
( s8 |- i* q, t5 B1 gQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
6 Y7 J% Q0 T- R" V5 e, G3 v- L[
    <$.1^4 + 1, 1>
5 E6 C; Q7 _4 R4 s]
$ t( R& M) ~$ O
R.<x> = QQ[]
- y$ |2 K& h+ `1 p  I$ {F6 = factor(x^6 - 1)
6 r/ S' e+ }0 V& U* N! c1 `F6

7 c6 a4 e; [7 s$ V2 Y' f. H5 s(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
3 U$ K( P1 N6 u! b) u/ S
, A6 g. w# V) R9 w" D' l0 lQ<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
F;
6 n/ j% \7 n; mD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
$ ?5 g3 {+ K; h* K1 i5 LCyclotomic Field of order 6 and degree 2
' O! d9 F* n9 H+ h, K2 j7 r( P$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
% I1 [! [9 U/ {4 }; }/ }% P    <$.1^2 - $.1 + 1, 1>
& q& N# F- s# H]

7 k4 y: G/ S% D. TR.<x> = QQ[]
- n* @/ K3 D( ~9 WF5 = factor(x^10 - 1)
F5
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
4 P* v# ?8 |- B9 U- i# u1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
+ O! Q% M/ N" _
Q<x> := QuadraticField(10);Q;
/ r) i0 g/ _4 s3 fC:=CyclotomicField(10);C;
4 b5 ]+ k5 i7 _& U% nFF:=CyclotomicPolynomial(10);FF;
7 r+ f0 _; w1 ~! G% f; e# j
F := QuadraticField(10);
3 I$ [1 Q; N  j& a! B1 @; HF;
D:=Factorization(FF) ;D;
9 ?' d; t8 Y4 g+ [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
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
) q! p, ^7 r2 r7 y: q" @1 z! j7 T[
# Z! v, K( _7 V6 @. C! I! h$ F    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
& C  [( S; w$ V]