# integral basis

Let $K$ be a number field. A set of algebraic integers $\{\alpha_{1},\ldots,\alpha_{s}\}$ is said to be an integral basis for $K$ if every $\gamma$ in $\mathcal{O}_{K}$ can be represented uniquely as an integer linear combination of $\{\alpha_{1},\ldots,\alpha_{s}\}$ (i.e. one can write $\gamma=m_{1}\alpha_{1}+\cdots+m_{s}\alpha_{s}$ with $m_{1},\ldots,m_{s}$ (rational) integers).

If $I$ is an ideal of $\mathcal{O}_{K}$, then $\{\alpha_{1},\ldots,\alpha_{s}\}\in I$ is said to be an integral basis for $I$ if every element of $I$ can be represented uniquely as an integer linear combination of $\{\alpha_{1},\ldots,\alpha_{s}\}$.

(In the above, $\mathcal{O}_{K}$ denotes the ring of algebraic integers of $K$.)

An integral basis for $K$ over $\mathbb{Q}$ is a basis for $K$ over $\mathbb{Q}$.

 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