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Mathematica > Mathematics and Algorithms > Number Theory > Algebraic Number Theory >

Built-in Mathematica Symbol

ToNumberField

]
expresses the algebraic number a in the number field generated by

ToNumberField[{a₁, a₂, ...},

]
expresses the a_i in the field generated by

ToNumberField[{a₁, a₂, ...}]
expresses the a_i in a common extension field generated by a single algebraic number.

ToNumberField gives AlgebraicNumber objects corresponding to elements of the rational extension .

The a_i and can be given in terms of Root or AlgebraicNumber objects, or ordinary rationals and radicals.

If is an algebraic integer the results will always be given in terms of AlgebraicNumber[, 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.

Express

in the number field generated by 2^1/4:

Express

in the number field generated by 2^1/4:

The generator

of the number field will autoreduce to an algebraic integer:

Radical expressions:

Root objects:

Express

and

in a common extension field:

Express algebraic numbers in the smallest common extension field:

Find a primitive element for

over

Convert an algebraic number to an explicit AlgebraicNumber object: