论坛 › 数论 › 虚二次域例两（-5/50）

lilianjie 发表于 2012-1-4 17:41

虚二次域例两（-5/50）

本帖最后由 lilianjie 于 2012-1-4 17:54 编辑

Q5:=QuadraticField(-5) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=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;
IsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+5);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

Name(M, -5);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -5) ;
NormEquation(M, -5) ;
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Univariate 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 + 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 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 - Q5.1, 1>,
    <w + Q5.1, 1>
]
-20

>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant


>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

20

>> Name(M, -5);
       ^
Runtime error in 'Name': Argument 2 (-5) should be in the range

1
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M
2
2
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M given by a rule
2
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
false
false
==============


Q5:=QuadraticField(-50) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=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;
IsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+50);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

Name(M, -50);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

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

Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Univariate 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
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
[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
-8

>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant


>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

8

>> Name(M, -50);
       ^
Runtime error in 'Name': Argument 2 (-50) should be in the range

1
Abelian 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
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]
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule
false
false

lilianjie 发表于 2012-1-4 17:51

看看-1.-3的两种：

Q5:=QuadraticField(-1) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=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;
IsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+1);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

Name(M, -1);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -1) ;
NormEquation(M, -1) ;

Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Univariate 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 + 1 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 1 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
[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
-4

>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant


>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

4

>> Name(M, -1);
       ^
Runtime error in 'Name': Argument 2 (-1) should be in the range

1
Abelian 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
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]
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 given by a rule
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 given by a rule
false
false
===============

Q5:=QuadraticField(-3) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=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;
IsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+3);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

Name(M, -3);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -3) ;
NormEquation(M, -3) ;

Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
Maximal Order of Q5
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 3 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
[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
-3

>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant


>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

3

>> Name(M, -3);
       ^
Runtime error in 'Name': Argument 2 (-3) should be in the range

1
Abelian 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
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]
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule
false
false

孤寂冷逍遥 发表于 2012-1-5 08:36

{:3_41:}{:3_41:}{:3_41:}{:3_41:}{:3_41:}

lilianjie 发表于 2012-1-5 13:02

本帖最后由 lilianjie 于 2012-1-5 15:20 编辑

Dirichlet character
Dirichlet class number formula

虚二次域复点1，实点为0，实二次域复点0，实点为2，实二次域只有两单位根

-1时，4个单位根1，-1, i, -i,w=4,            N=4，互素（1，3），    (Z/4Z)*------->C*      χ（1mod4）=1，χ（3mod4）=-1，h=4/（2*4）*Σ=1

-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
h=-6/（2*3）*Σ=1

-5时  2个单位根                              N=20   N=3  互素（1，3，7,9,11,13,17,19），    (Z/5Z)*------->C*         χ（1mod20）=1，χ（3mod20）=-1， χ（7mod20）=1， χ（9mod20）=1，χ（11mod20）=1，χ（13mod20）=1，χ（17mod20）=1，χ（19mod20）=1，



h=2/（2*20）*Σ=2



-50时  个单位根                          N=200

lilianjie1 发表于 2012-1-5 13:51

Dirichlet character

lilianjie1 发表于 2012-1-5 20:37

pell   equation

lilianjie 发表于 2012-1-9 20:28

本帖最后由 lilianjie 于 2012-1-9 20:30 编辑

F := QuadraticField(NextPrime(5));

KK := QuadraticField(7);KK;
K:=MaximalOrder(KK);
Conductor(KK);
ClassGroup(KK) ;
QuadraticClassGroupTwoPart(KK) ;
NormEquation(F, 7);
A:=K!7;A;
B:=K!14;B;
Discriminant(KK)

Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
28
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of K
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
inverse]
false
7
14
28

lilianjie 发表于 2012-1-9 20:44

本帖最后由 lilianjie 于 2012-1-10 11:23 编辑







分圆域：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);
C:=CyclotomicField(6);C;
CyclotomicPolynomial(6);
CC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
MinimalField(CC!7) ;
MinimalField(CC!8) ;
MinimalField(CC!9) ;
MinimalCyclotomicField(CC!7) ;
RootOfUnity(11);RootOfUnity(111);
Minimise(CC!123);
Conductor(CC) ;
CyclotomicOrder(CC) ;

CyclotomicAutomorphismGroup(CC) ;

Cyclotomic Field of order 5 and degree 4
$.1^4 + $.1^3 + $.1^2 + $.1 + 1
Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
Cyclotomic Field of order 7 and degree 6
$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
Rational Field
Rational Field
Rational Field
zeta_11
zeta_111
123
7
7
Permutation group acting on a set of cardinality 6
Order = 6 = 2 * 3
    (1, 2)(3, 5)(4, 6)
    (1, 3, 6, 2, 5, 4)
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC
Composition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
Degree 6, Order 2 * 3 and
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC

lilianjie 发表于 2012-1-10 11:34

本帖最后由 lilianjie 于 2012-1-10 17:41 编辑

lilianjie 发表于 2012-1-9 20:44 static/image/common/back.gif
分圆域：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);


分圆域：
分圆域：123

R.<x> = Q[]
F8 = factor(x^8 - 1)
F8

(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;
FF:=CyclotomicPolynomial(8);FF;

F := QuadraticField(8);
F;
D:=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
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
    <$.1^4 + 1, 1>
]

R.<x> = QQ[]
F6 = factor(x^6 - 1)
F6

(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

Q<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
F;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
    <$.1^2 - $.1 + 1, 1>
]

R.<x> = QQ[]
F5 = 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)

Q<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

F := QuadraticField(10);
F;
D:=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
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

lilianjie1 发表于 2012-1-10 11:55

分圆域：123

查看完整版本: 虚二次域例两（-5/50）