# algebraic integer

Let $K$ be an extension (http://planetmath.org/ExtensionField) of $\mathbb{Q}$ contained in $\mathbb{C}$. A number $\alpha\in K$ is called an algebraic integer of $K$ if it is the root of a monic polynomial with coefficients in $\mathbb{Z}$, i.e., an element of $K$ that is integral over $\mathbb{Z}$. Every algebraic integer is an algebraic number (with $K=\mathbb{C}$), but the converse is false.

 Title algebraic integer Canonical name AlgebraicInteger Date of creation 2013-03-22 11:45:41 Last modified on 2013-03-22 11:45:41 Owner KimJ (5) Last modified by KimJ (5) Numerical id 13 Author KimJ (5) Entry type Definition Classification msc 11R04 Classification msc 62-01 Classification msc 03-01 Related topic IntegralBasis Related topic CyclotomicUnitsAreAlgebraicUnits Related topic FundamentalUnits Related topic Monic2 Related topic RingWithoutIrreducibles