ToNumberField

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`ToNumberField`

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ToNumberField[a,θ]

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

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ToNumberField[{a₁,a₂,…},θ]

expresses the a_i in the field generated by θ.

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ToNumberField[{a₁,a₂,…}]

expresses the a_i 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 a_i 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 a_i 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.
ToNumberField[x] converts any form of algebraic number to an explicit AlgebraicNumber object.

Examples

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Basic Examples (1)Summary of the most common use cases

Express in the number field generated by :

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https://wolfram.com/xid/0dqu615p7ou-bj5mzu

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Scope (6)Survey of the scope of standard use cases

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

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https://wolfram.com/xid/0dqu615p7ou-zz4wfz

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Radical expressions:

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https://wolfram.com/xid/0dqu615p7ou-055jhw

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Root objects:

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https://wolfram.com/xid/0dqu615p7ou-u9s0c9

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AlgebraicNumber objects:

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https://wolfram.com/xid/0dqu615p7ou-p27ovy

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Express and in a common extension field:

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https://wolfram.com/xid/0dqu615p7ou-bf5ay7

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Express algebraic numbers in the smallest common extension field:

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https://wolfram.com/xid/0dqu615p7ou-otlpzs

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Applications (1)Sample problems that can be solved with this function

Find a primitive element for over :

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https://wolfram.com/xid/0dqu615p7ou-5fuyhq

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Properties & Relations (1)Properties of the function, and connections to other functions

Convert an algebraic number to an explicit AlgebraicNumber object:

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https://wolfram.com/xid/0dqu615p7ou-b1ei1

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