# Extension

is an option for various polynomial and algebraic functions that specifies generators for the algebraic number field to be used.

# Details • For polynomial functions, Extension determines the algebraic number field in which the coefficients are assumed to lie.
• The setting Extension->a specifies the field consisting of the rationals extended by the algebraic number a.
• Extension->{a1,a2,} specifies the field .
• The ai must be exact numbers, and can involve radicals as well as Root and AlgebraicNumber objects.
• specifies that any algebraic numbers that appear in the input should be included in the extension field.
• For polynomial functions, the default setting specifies that all coefficients are required to be rational. Any algebraic numbers appearing in input are treated like independent variables.
• Extension->{a1,a2,} includes both the ai and any algebraic numbers in the input.
• is equivalent to .

# Examples

open allclose all

## Basic Examples(2)

Factor a polynomial over :

PolynomialGCD over the field generated by the algebraic numbers present in the coefficients:

## Scope(8)

By default, factorization is performed over the rationals:

This specifies the factorization should be done over the rationals extended by :

Here the factorization is done over the rationals extended by and I:

By default, PolynomialGCD treats algebraic numbers as independent variables:

This computes the GCD over the algebraic number field generated by the coefficients:

By default, Together treats algebraic numbers as independent variables:

With , Together recognizes algebraically dependent coefficients:

By default, the norm is computed in the field generated by the AlgebraicNumber object:

This computes the norm in the field in which the AlgebraicNumber object is represented:

This computes the norm in the field generated by :

## Properties & Relations(1)

For Factor, is equivalent to :

Introduced in 1996
(3.0)
|
Updated in 2007
(6.0)