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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, number, contained
There are 38 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 6196 times total.

Classification:
AMS MSC11R04 (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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