数字字段¶
分支¶
在Sage中如何计算具有给定判别和分支的数字域?
Sage可以访问Jones数据库中分枝数有界且度小于或等于6的数域。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
列出数据库中所有分支最多为2的字段的度和判别度:
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)]
列出精确2度字段的判别式,在2之外未赋值:
sage: [k.disc() for k in J.unramified_outside([2],2)] # optional - database
[8, -4, -8]
列出数据库中立方字段的判别式,精确到3和5:
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
列出数据库中101处分支的所有字段:
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]
班级编号¶
在Sage中,如何计算数字字段的类号?
这个 class_number
是与方形场对象相关联的方法:
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
下面是一个涉及更一般类型的数字字段的示例:
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
另请参阅http://mathworld.wolfram.com/ClassNumber.html在Math World网站上获取表格、公式和背景信息。
对于分圆域,请尝试:
sage: K = CyclotomicField(19) sage: K.class_number() # long time 1
有关详细信息,请参阅中的文档字符串 ring/number_field.py
文件。
积分基¶
在Sage中如何计算一个数域的整数基?
Sage可以计算这个数字域的元素列表,这些元素是数字域的整数环的基础。
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]
接下来我们计算一个立方域的整数环,其中2是一个“基本判别因子”,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]