integral basis

Let $K$ be a number field. A set of algebraic integers $\{{\alpha }_1,\mathrm{\dots },{\alpha }_s\}$ is said to be an integral basis for $K$ if every $\gamma$ in ${𝒪}_K$ can be represented uniquely as an integer linear combination of $\{{\alpha }_1,\mathrm{\dots },{\alpha }_s\}$ (i.e. one can write $\gamma =m_1{\alpha }_1+\mathrm{\cdots }+m_s{\alpha }_s$ with $m_1,\mathrm{\dots },m_s$ (rational) integers).

If $I$ is an ideal of ${𝒪}_K$, then $\{{\alpha }_1,\mathrm{\dots },{\alpha }_s\}\in I$ is said to be an integral basis for $I$ if every element of $I$ can be represented uniquely as an integer linear combination of $\{{\alpha }_1,\mathrm{\dots },{\alpha }_s\}$.

(In the above, ${𝒪}_K$ 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