13.2 Number Field Elements

Module: sage.rings.number_field.number_field_element

Module-level Functions

is_NumberFieldElement( x)

Class: NumberFieldElement

class NumberFieldElement
An element of a number field.
NumberFieldElement( self, parent, f)

INPUT:
    parent -- a number field
    f -- defines an element of a number field.

The following examples illustrate creation of elements of number fields, and some basic arithmetic.

First we define a polynomial over Q.

sage: x = PolynomialRing(Q).0
sage: f = x^2 + 1

Next we use f to define the number field.

sage: K = NumberField(f, "a"); K
Number Field in a with defining polynomial x^2 + 1
sage: a = K.gen()
sage: a^2
-1
sage: (a+1)^2
2*a
sage: a^2
-1
sage: z = K(5); 1/z
1/5

We create a cube root of 2.

sage: K = NumberField(x^3 - 2, "b")
sage: b = K.gen()
sage: b^3
2
sage: (b^2 + b + 1)^3
12*b^2 + 15*b + 19

This example illustrates save and load:

sage: K, a = NumberField(x^17 - 2, 'a').objgen()
sage: s = a^15 - 19*a + 3
sage: loads(s.dumps()) == s
True

Functions: charpoly,$  $ list,$  $ matrix,$  $ minpoly,$  $ multiplicative_order,$  $ norm,$  $ polynomial,$  $ trace

matrix( self)

The matrix of right multiplication by the element on the power basis $ 1, x, x^2, \ldots, x^{d-1}$ for the number field. Thus the rows of this matrix give the images of each of the $ x^i$ .

Special Functions: __cmp__,$  $ __getitem__,$  $ __int__,$  $ __invert__,$  $ __long__,$  $ __neg__,$  $ __pow__,$  $ __repr__,$  $ _add_,$  $ _div_,$  $ _im_gens_,$  $ _integer_,$  $ _latex_,$  $ _mul_,$  $ _pari_,$  $ _pari_init_,$  $ _rational_,$  $ _set_multiplicative_order,$  $ _sub_

_mul_( self, other)

Returns the product of self and other as elements of a number field.

_pari_( self, [var=None])

Return PARI C-library object representation of self.

_pari_init_( self, [var=None])

Return GP/PARI string representation of self.

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