数域

分歧 (Ramification)

如何在 Sage 中计算具有给定判别式和分歧的数域？

Sage 可以访问 Jones 数域数据库，该数据库包含有界分歧且度数不超过 6 的数域。 该数据库必须单独安装（database_jones_numfield）。

首先加载数据库：

sage: J = JonesDatabase()            # optional - database
sage: J                              # optional - database
John Jones's table of number fields with bounded ramification and degree <= 6

>>> from sage.all import *
>>> J = JonesDatabase()            # optional - database
>>> J                              # optional - database
John Jones's table of number fields with bounded ramification and degree <= 6

列出数据库中所有分歧最多为 2 的数域的度数和判别式：

sage: [(k.degree(), k.disc()) for k in J.unramified_outside([2])] # optional - database
[(4, -2048), (2, 8), (4, -1024), (1, 1), (4, 256), (2, -4), (4, 2048), (4, 512), (4, 2048), (2, -8), (4, 2048)]

>>> from sage.all import *
>>> [(k.degree(), k.disc()) for k in J.unramified_outside([Integer(2)])] # optional - database
[(4, -2048), (2, 8), (4, -1024), (1, 1), (4, 256), (2, -4), (4, 2048), (4, 512), (4, 2048), (2, -8), (4, 2048)]

列出在 2 之外无分歧且度数恰好为 2 的域的判别式：

sage: [k.disc() for k in J.unramified_outside([2],2)] # optional - database
[8, -4, -8]

>>> from sage.all import *
>>> [k.disc() for k in J.unramified_outside([Integer(2)],Integer(2))] # optional - database
[8, -4, -8]

列出数据库中在 3 和 5 处有分歧的立方域的判别式：

sage: [k.disc() for k in J.ramified_at([3,5],3)] # optional - database
[-6075, -6075, -675, -135]
sage: factor(6075)
3^5 * 5^2
sage: factor(675)
3^3 * 5^2
sage: factor(135)
3^3 * 5

>>> from sage.all import *
>>> [k.disc() for k in J.ramified_at([Integer(3),Integer(5)],Integer(3))] # optional - database
[-6075, -6075, -675, -135]
>>> factor(Integer(6075))
3^5 * 5^2
>>> factor(Integer(675))
3^3 * 5^2
>>> factor(Integer(135))
3^3 * 5

列出所有在 101 处有分歧的域：

sage: J.ramified_at(101)                     # optional - database
[Number Field in a with defining polynomial x^2 - 101,
 Number Field in a with defining polynomial x^4 - x^3 + 13*x^2 - 19*x + 361,
 Number Field in a with defining polynomial x^5 - x^4 - 40*x^3 - 93*x^2 - 21*x + 17,
 Number Field in a with defining polynomial x^5 + x^4 - 6*x^3 - x^2 + 18*x + 4,
 Number Field in a with defining polynomial x^5 + 2*x^4 + 7*x^3 + 4*x^2 + 11*x - 6]

>>> from sage.all import *
>>> J.ramified_at(Integer(101))                     # optional - database
[Number Field in a with defining polynomial x^2 - 101,
 Number Field in a with defining polynomial x^4 - x^3 + 13*x^2 - 19*x + 361,
 Number Field in a with defining polynomial x^5 - x^4 - 40*x^3 - 93*x^2 - 21*x + 17,
 Number Field in a with defining polynomial x^5 + x^4 - 6*x^3 - x^2 + 18*x + 4,
 Number Field in a with defining polynomial x^5 + 2*x^4 + 7*x^3 + 4*x^2 + 11*x - 6]

类数 (Class numbers)

如何在 Sage 中计算数域的类数 (class number)？

class_number 是一个与 QuadraticField 对象相关的方法：

sage: K = QuadraticField(29, 'x')
sage: K.class_number()
1
sage: K = QuadraticField(65, 'x')
sage: K.class_number()
2
sage: K = QuadraticField(-11, 'x')
sage: K.class_number()
1
sage: K = QuadraticField(-15, 'x')
sage: K.class_number()
2
sage: K.class_group()
Class group of order 2 with structure C2 of Number Field in x with defining polynomial x^2 + 15 with x = 3.872983346207417?*I
sage: K = QuadraticField(401, 'x')
sage: K.class_group()
Class group of order 5 with structure C5 of Number Field in x with defining polynomial x^2 - 401 with x = 20.02498439450079?
sage: K.class_number()
5
sage: K.discriminant()
401
sage: K = QuadraticField(-479, 'x')
sage: K.class_group()
Class group of order 25 with structure C25 of Number Field in x with defining polynomial x^2 + 479 with x = 21.88606862823929?*I
sage: K.class_number()
25
sage: K.pari_polynomial()
x^2 + 479
sage: K.degree()
2

>>> from sage.all import *
>>> K = QuadraticField(Integer(29), 'x')
>>> K.class_number()
1
>>> K = QuadraticField(Integer(65), 'x')
>>> K.class_number()
2
>>> K = QuadraticField(-Integer(11), 'x')
>>> K.class_number()
1
>>> K = QuadraticField(-Integer(15), 'x')
>>> K.class_number()
2
>>> K.class_group()
Class group of order 2 with structure C2 of Number Field in x with defining polynomial x^2 + 15 with x = 3.872983346207417?*I
>>> K = QuadraticField(Integer(401), 'x')
>>> K.class_group()
Class group of order 5 with structure C5 of Number Field in x with defining polynomial x^2 - 401 with x = 20.02498439450079?
>>> K.class_number()
5
>>> K.discriminant()
401
>>> K = QuadraticField(-Integer(479), 'x')
>>> K.class_group()
Class group of order 25 with structure C25 of Number Field in x with defining polynomial x^2 + 479 with x = 21.88606862823929?*I
>>> K.class_number()
25
>>> K.pari_polynomial()
x^2 + 479
>>> K.degree()
2

下面是一个更为一般的数域类型的例子：

sage: x = PolynomialRing(QQ, 'x').gen()
sage: K = NumberField(x^5+10*x+1, 'a')
sage: K
Number Field in a with defining polynomial x^5 + 10*x + 1
sage: K.degree()
5
sage: K.pari_polynomial()
x^5 + 10*x + 1
sage: K.discriminant()
25603125
sage: K.class_group()
Class group of order 1 of Number Field in a with defining
polynomial x^5 + 10*x + 1
sage: K.class_number()
1

>>> from sage.all import *
>>> x = PolynomialRing(QQ, 'x').gen()
>>> K = NumberField(x**Integer(5)+Integer(10)*x+Integer(1), 'a')
>>> K
Number Field in a with defining polynomial x^5 + 10*x + 1
>>> K.degree()
5
>>> K.pari_polynomial()
x^5 + 10*x + 1
>>> K.discriminant()
25603125
>>> K.class_group()
Class group of order 1 of Number Field in a with defining
polynomial x^5 + 10*x + 1
>>> K.class_number()
1

  • 对于循环域，可以尝试：

    sage: K = CyclotomicField(19)
sage: K.class_number()    # long time
1

    >>> from sage.all import *
>>> K = CyclotomicField(Integer(19))
>>> K.class_number()    # long time
1

更多详情，请参见 ring/number_field.py 中的文档。

整基 (Integral basis)

如何在 Sage 中计算数域的整基？

Sage 可以计算数域的元素列表，该列表是该数域的整数全环的基。

sage: x = PolynomialRing(QQ, 'x').gen()
sage: K = NumberField(x^5+10*x+1, 'a')
sage: K.integral_basis()
[1, a, a^2, a^3, a^4]

>>> from sage.all import *
>>> x = PolynomialRing(QQ, 'x').gen()
>>> K = NumberField(x**Integer(5)+Integer(10)*x+Integer(1), 'a')
>>> K.integral_basis()
[1, a, a^2, a^3, a^4]

接下来我们计算立方域的整数环，其中 2 是“基本判别式因子”，因此整数环不是由单个元素生成的。

sage: x = PolynomialRing(QQ, 'x').gen()
sage: K = NumberField(x^3 + x^2 - 2*x + 8, 'a')
sage: K.integral_basis()
[1, 1/2*a^2 + 1/2*a, a^2]

>>> from sage.all import *
>>> x = PolynomialRing(QQ, 'x').gen()
>>> K = NumberField(x**Integer(3) + x**Integer(2) - Integer(2)*x + Integer(8), 'a')
>>> K.integral_basis()
[1, 1/2*a^2 + 1/2*a, a^2]