实二次域(5/50)例2

lilianjie

* V1 j; s% ^; \1 c1 }* A. M
! a$ B6 y( O- Y4 y8 bQ5:=QuadraticField(5) ;
Q5;
Q<w> :=PolynomialRing(Q5);Q;

EquationOrder(Q5);
/ S; u# L4 [+ b* Z2 ZM:=MaximalOrder(Q5) ;
M;
, R" Z; a) A; b4 T) v: _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);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);
Factorization(w^2-3);
Discriminant(Q5) ;
$ ]+ Q& d6 t0 |1 G+ g- ^, n& R) I5 }FundamentalUnit(Q5) ;
FundamentalUnit(M);
- O: l- e# ?! C, v! f0 |5 KConductor(Q5) ;
Name(Q5, 1);
Name(M, 1);
& E4 s/ D+ ^7 G% `Conductor(M);
/ q' y. P8 m, SClassGroup(Q5) ;
ClassGroup(M);
' X' W5 _2 z$ f: x1 d& YClassNumber(Q5) ;
ClassNumber(M) ;
9 E: }6 h5 x( K  l
PicardGroup(M) ;
PicardNumber(M) ;

( B9 H5 m4 ]8 D2 i% d' p' _
! |) ]# [3 H" XQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);

9 {% l; u& B- Y
% x9 d5 L" ^% n4 y5 U, tNormEquation(Q5, 5) ;
  d/ B- O6 W0 N1 f, a% cNormEquation(M, 5) ;
( |( L7 y6 ]% X8 F; {# Y
! |5 v) I$ u+ E
' U7 C3 P7 b6 j: ]! Y; l  \" gQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
) x8 @. ?) t9 H3 X4 l& z- bUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
5 G8 A& R$ |! l) C: G4 K/ cMaximal Order of Q5
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
- h0 d- P  {% C  P/ a  ttrue Order of conductor 16 in Q5
% p& y, c' [; j# d1 ]true Order of conductor 625 in Q5
7 P5 m6 B% o3 f( J6 j6 Ytrue Order of conductor 391736900121876544 in Q5
0 Y$ U' p3 g: B[
    <w^2 - 3, 1>
/ {$ r1 ]/ n4 t6 I0 g6 U7 |. L]
: X7 I" F% b* Z5 M5
1/2*(-Q5.1 + 1)
-$.2 + 1
, |6 f  i0 W4 [) l5
7 B4 A( i/ H+ K8 F- v( _; VQ5.1
7 W. l6 c1 |, f$ w$.2
1
Abelian Group of order 1
4 p# {5 z+ Z8 B1 KMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
. E6 r. x' _- B' i% z, [  |1
& t2 {4 O1 P7 t! `! hAbelian Group of order 1
3 ]5 S( o  ?) BMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
. h% @- p7 t  i8 d) linverse]
1
( C# V" d5 v, z  o  LAbelian Group of order 1
! c# R2 @2 F0 b) P" ^/ NMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
( |6 j$ {3 P% v) v% ]Abelian Group of order 1
3 K7 r- H% L* t6 a) qMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
  {" e; ?- P% s) `( C2 J# Dtrue [ 1/2*(Q5.1 + 5) ]
6 a' W3 R- ^' M$ L$ t/ H7 s$ W1 utrue [ -2*$.2 + 1 ]
7 D# {7 p: F# q/ n. p/ \! Q
1 Q, B; K$ F* k6 ?, d& Y
& k2 Z, {1 y; q; b: ~6 s1 |8 H, w( r


+ [& f7 M: |, Q& I
, @1 k+ {- q' n# i' N% e! T

) O* R6 e+ N8 W2 P

& f" ^% a7 v) P- P4 e
. i, y  ^+ t6 u7 Q5 m==============
  B5 _6 m6 W5 K) n
6 J8 Q) g  v( Y: qQ5:=QuadraticField(50) ;
" ^9 P5 T# M: WQ5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
7 E2 V9 x% e0 k8 q$ C9 ]. q1 J. QM:=MaximalOrder(Q5) ;
M;
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;
% @" r# u* K/ a& rIsQuadratic(Q5);
3 P. z. J0 a" Q! b- DIsQuadratic(S1);
* \8 ]5 ?. c+ f6 k4 gIsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2-50);  
5 G4 m7 u4 B8 H! {& uDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

( u& R6 z8 [. L7 y. IName(M, 50);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
6 `8 ]& y( Z0 m# C+ ZClassNumber(Q5) ;
; ~$ K4 v- O# KClassNumber(M) ;
PicardGroup(M) ;
% A$ u' U! P9 B# M( ?0 R/ G& R' h# WPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
7 \8 V* Q4 n5 B4 x0 ?) v2 E9 qQuadraticClassGroupTwoPart(M);
NormEquation(Q5, 50) ;
/ }% c  f" O# b: b, j5 N' [* _2 u, YNormEquation(M, 50) ;
' p  x: c7 _9 }0 q- C2 s1 s6 A
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
, m6 D/ {; W1 c( E6 {: {5 ~# fEquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
3 E  ?9 O2 p8 l, LOrder of conductor 625888888 in Q5
" m$ {0 I4 z- j) n; ^, vtrue Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
) J) `. p# y; p: J' P$ F8 P4 itrue Order of conductor 1 in Q5
7 x/ q$ L0 g3 T7 {true Order of conductor 1 in Q5
[
4 z6 v5 m) I5 A. ~    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
1 y; R% `* S& R2 a; ^8
0 E4 m/ m0 ?& K) h" |Q5.1 + 1
$.2 + 1
8
2 \: [. \% K+ g( [+ m
>> Name(M, 50);
4 r7 k. v, `5 n% p       ^
& y. A+ O4 G3 j8 L. D: `Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]

/ B3 H/ Y+ q9 p& d, e3 {6 \7 ~6 U+ c1
! R) ?+ F3 ^7 V7 O. P' e. O# B! ~Abelian Group of order 1
& A7 b0 f' F& j4 E" l1 Q( s% MMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
8 G6 J6 D# ]: i/ }Mapping from: Abelian Group of order 1 to Set of ideals of M
1
7 w5 r. l/ r7 N2 g  o6 Y/ {1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
' D# e2 u+ t8 o5 n4 u, qinverse]
- o: \$ c) `& Q1
Abelian Group of order 1
; C" X% y/ p  L( @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
8 given by a rule [no inverse]
. s  x6 l8 r2 ^9 j/ J# Y/ @true [ 5*Q5.1 + 10 ]
  y/ W# o& n% G# [/ Ftrue [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

) z9 G* S  u2 y% A
% A7 m! U* J" \8 Y& c$ X基本单位计算fundamentalunit ：
5 mod4 =1                                              50 mod 4=2
1 _' u" a- y2 j8 b* m
x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.
! c: C9 A% j( G5 Z" Z! n8 w

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
, U3 w4 Z* B4 w" C$ c2 T; t; l基本单位fundamentalunit ：
5 mod4 =1                              50 mod 4=2

4 }7 U  \& v3 z' p! a基本单位fundamentalunit

lilianjie1

判别式计算Discriminant
. \: a; r, J4 }& T' r$ Z3 t! H9 }
5MOD 4=1
; B& H6 Z% J8 c1 d$ C* X7 A
（1+1）/2=1          （1-1）/2=0
2 d) V- w5 v+ O/ Y. L
D=5


50MOD 4=2
! r0 R  S0 D) l7 K9 k' C" f$ XD=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44
- h0 l/ w6 K5 B' n! {+ ^
' C' R7 ^, y! K3 G# Q3 F- A8 n分圆多项式总是原多项式因子：
% U- m" L* g4 M- Y6 }8 YC:=CyclotomicField(5);C;
5 h' F% R- |0 K5 o6 sCyclotomicPolynomial(5);

分圆域：
分圆域：123
) V3 \: U, A1 F  P
R.<x> = Q[]
F8 = factor(x^8 - 1)
F8

7 i. A1 Y) D( z. q" z( ~(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
6 n; ~: E- }/ d
Q<x> := QuadraticField(8);Q;
/ L2 h! I2 z5 ?5 TC:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;
! n* C2 D% d5 F  k5 O, N2 w+ }3 N
$ L& a1 e$ `3 D+ {F := QuadraticField(8);
F;
) g7 F$ `" N4 m1 a1 p; T. `  aD:=Factorization(FF) ;D;
4 T1 Y; M) g( R3 L% WQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
; I& a: `$ i' u; {% g& ~% gCyclotomic Field of order 8 and degree 4
$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
  [* K! o' `6 H! I    <$.1^4 + 1, 1>
]
3 p$ G9 `; _4 P2 _
4 k) h7 k8 Y- f* IR.<x> = QQ[]
6 _, Y3 j$ `, ]3 NF6 = factor(x^6 - 1)
F6
+ p0 B1 x" V) u- ~0 Z0 O* L, x
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
' h3 w) I4 e9 P8 h8 G  ?; V- t
Q<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
) E+ {8 n: L5 o/ O, @FF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
' C  }( L9 _. q9 t( M# vF;
4 `8 x- N4 B& G: t( c# PD:=Factorization(FF) ;D;
. Z1 k7 X2 j4 n3 F* Z# l& ^Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
, u2 \3 s" M$ |$ E- r) VCyclotomic Field of order 6 and degree 2
5 G9 l6 U' v0 S! {& }3 R- o7 N# ~$.1^2 - $.1 + 1
# R8 H3 O0 t+ a# a! IQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
9 ?9 `$ ?+ j) O/ I[
    <$.1^2 - $.1 + 1, 1>
]
8 g- c$ n" U( u# d$ o3 M3 b. j' g/ Q
R.<x> = QQ[]
F5 = factor(x^10 - 1)
% Z; c& h6 X9 L* N; r8 ~: mF5
(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)
2 U8 `# k: \/ z! B- N& h0 h
7 c2 ?  n; k1 A  ]8 r' zQ<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
4 E/ f2 G: [  cFF:=CyclotomicPolynomial(10);FF;
: w1 I! ?8 ^# q/ @1 E2 x& o! p/ |
# `, v7 M9 f" |3 A( NF := QuadraticField(10);
F;
7 z2 E: t+ ~' R6 L( |$ f+ z  y9 YD:=Factorization(FF) ;D;
6 T; J  ~' V/ g7 w1 }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 $.1^2 - 10 over the Rational Field
[
5 j. U! U1 T4 `    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]