# 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 :

Wolfram Research (1996), Extension, Wolfram Language function, https://reference.wolfram.com/language/ref/Extension.html (updated 2007).

#### Text

Wolfram Research (1996), Extension, Wolfram Language function, https://reference.wolfram.com/language/ref/Extension.html (updated 2007).

#### CMS

Wolfram Language. 1996. "Extension." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/Extension.html.

#### APA

Wolfram Language. (1996). Extension. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Extension.html

#### BibTeX

@misc{reference.wolfram_2023_extension, author="Wolfram Research", title="{Extension}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/Extension.html}", note=[Accessed: 30-September-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2023_extension, organization={Wolfram Research}, title={Extension}, year={2007}, url={https://reference.wolfram.com/language/ref/Extension.html}, note=[Accessed: 30-September-2023 ]}