gives an integral basis for the field 
generated by the algebraic number 
.
NumberFieldIntegralBasis
gives an integral basis for the field 
generated by the algebraic number 
.
Details
- NumberFieldIntegralBasis gives a list of algebraic integers generating the module of integers in the field
. - The basis produced by NumberFieldIntegralBasis may not be unique.
Examples
open all close allBasic Examples (1)
Scope (4)
NumberFieldIntegralBasis[Sqrt[2] + Sqrt[3]]Root objects:
NumberFieldIntegralBasis[Root[2 + 3#1 ^ 3&, 2]]AlgebraicNumber objects:
NumberFieldIntegralBasis[AlgebraicNumber[Root[-10 + #1^3&, 1], {1, 0, 1 / 3}]]NumberFieldIntegralBasis threads over lists:
NumberFieldIntegralBasis[{Sqrt[2], Sqrt[5]}]Properties & Relations (3)
Elements of an integral basis are all algebraic integers:
NumberFieldIntegralBasis[E ^ (I Pi / 4)]AlgebraicIntegerQ /@ %Algebraic integers are linear combinations of elements of the integral basis:
basis = NumberFieldIntegralBasis[Sqrt[2 + Sqrt[3]]]basis.{-1, 2, 2, 3}AlgebraicIntegerQ[%]Use ToNumberField to find an integral basis for the field 
:
primitive = ToNumberField[{Sqrt[2], I}, All][[1, 1]]NumberFieldIntegralBasis[primitive]Text
Wolfram Research (2007), NumberFieldIntegralBasis, Wolfram Language function, https://reference.wolfram.com/language/ref/NumberFieldIntegralBasis.html.
CMS
Wolfram Language. 2007. "NumberFieldIntegralBasis." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NumberFieldIntegralBasis.html.
APA
Wolfram Language. (2007). NumberFieldIntegralBasis. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NumberFieldIntegralBasis.html