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.
- ToNumberField[x] converts any form of algebraic number to an explicit AlgebraicNumber object.
Examples
open all close allBasic Examples (1)
Scope (6)
The generator θ of the number field will autoreduce to an algebraic integer:
ToNumberField[2, 1 / 2]ToNumberField[2Sqrt[2] + 1, Sqrt[2] / 2]Root objects:
ToNumberField[Root[-25 - 24 #1 - 12 #1^2 + 4 #1^3&, 1], Root[-3 + 2#1^3&, 1]]AlgebraicNumber objects:
ToNumberField[E ^ (Pi I / 4), I AlgebraicNumber[Sqrt[2], {1, 2}]]Express 
and 
in a common extension field:
ToNumberField[{Sqrt[3], (1 + I Sqrt[3]) / 2}]Express algebraic numbers in the smallest common extension field:
ToNumberField[{AlgebraicNumber[Root[1 - 10 #1^2 + #1^4&, 4], {0, -9 / 2, 0, 1 / 2}], Sqrt[5]}, All]Applications (1)
Properties & Relations (1)
Convert an algebraic number to an explicit AlgebraicNumber object:
ToNumberField[Sqrt[Sqrt[2] + Sqrt[3]]]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