integral basis


Let K be a number field. A set of algebraic integersMathworldPlanetmath {α1,,αs} is said to be an integral basis for K if every γ in 𝒪K can be represented uniquely as an integer linear combinationMathworldPlanetmath of {α1,,αs} (i.e. one can write γ=m1α1++msαs with m1,,ms (rational) integers).

If I is an ideal of 𝒪K, then {α1,,αs}I is said to be an integral basis for I if every element of I can be represented uniquely as an integer linear combination of {α1,,αs}.

(In the above, 𝒪K denotes the ring of algebraic integers of K.)

An integral basis for K over is a basis for K over .

Title integral basis
Canonical name IntegralBasis
Date of creation 2013-03-22 12:36:03
Last modified on 2013-03-22 12:36:03
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 12
Author rspuzio (6075)
Entry type Definition
Classification msc 11R04
Synonym minimal basis
Synonym minimal bases
Related topic AlgebraicInteger
Related topic Integral
Related topic Basis
Related topic DiscriminantOfANumberField
Related topic ConditionForPowerBasis
Related topic BasisOfIdealInAlgebraicNumberField
Related topic CanonicalFormOfElementOfNumberField
Defines integral bases