# 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

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## 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: