# ToNumberField

ToNumberField[a,θ]

expresses the algebraic number a in the number field generated by θ.

ToNumberField[{a1,a2,},θ]

expresses the ai in the field generated by θ.

ToNumberField[{a1,a2,}]

expresses the ai in a common extension field generated by a single algebraic number.

# Details

• ToNumberField gives AlgebraicNumber objects corresponding to elements of the rational extension .
• ToNumberField[a,θ] remains unevaluated if a does not exist in .
• The ai and θ can be given in terms of Root or AlgebraicNumber objects, or ordinary rationals and radicals.
• If θ is an algebraic integer the results will always be given in terms of AlgebraicNumber[θ,list].
• ToNumberField[{a1,a2,}] gives a representation of the ai in terms of a primitive element of the field .
• ToNumberField[{a1,a2,}] is equivalent to ToNumberField[{a1,a2,},Automatic], and does not necessarily use the smallest common field extension.
• ToNumberField[{a1,a2,},All] always uses the smallest common field extension.
• converts any form of algebraic number to an explicit AlgebraicNumber object.

# Examples

open allclose all

## Basic Examples(1)

Express in the number field generated by :

## Scope(6)

The generator θ of the number field will autoreduce to an algebraic integer:

Root objects:

AlgebraicNumber objects:

Express and in a common extension field:

Express algebraic numbers in the smallest common extension field:

## Applications(1)

Find a primitive element for over :

## Properties & Relations(1)

Convert an algebraic number to an explicit AlgebraicNumber object:

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

#### Text

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

#### CMS

Wolfram Language. 2007. "ToNumberField." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ToNumberField.html.

#### APA

Wolfram Language. (2007). ToNumberField. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ToNumberField.html

#### BibTeX

@misc{reference.wolfram_2022_tonumberfield, author="Wolfram Research", title="{ToNumberField}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/ToNumberField.html}", note=[Accessed: 08-June-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_tonumberfield, organization={Wolfram Research}, title={ToNumberField}, year={2007}, url={https://reference.wolfram.com/language/ref/ToNumberField.html}, note=[Accessed: 08-June-2023 ]}