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algebraic integer (Definition)

Let $ K$ be an extension 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.



"algebraic integer" is owned by KimJ.
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See Also: integral basis, the cyclotomic units are algebraic units, fundamental units, monic, ring without irreducibles

Keywords:  algebraic number theory

Attachments:
ideal norm (Definition) by pahio
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Cross-references: converse, algebraic number, integral, coefficients, monic polynomial, root, contained
There are 35 references to this entry.

This is version 8 of algebraic integer, born on 2001-10-15, modified 2006-10-04.
Object id is 210, canonical name is AlgebraicInteger.
Accessed 5006 times total.

Classification:
AMS MSC: 11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)
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may wish to generalize by saforres on 2002-04-23 21:26:05
You might want to add the simple generalization to an extension of the rationals that isn't all of C:

Let K be an extension of Q. A number $\alpha \in K$ is called an algebraic integer of K it is the root of a monic polynomial with coefficients in $\mathbb{Z}$.
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