Talk:Algebraic integer

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The definition of algebraic integer as defined in the article does not satisfy many desirable properties of integers, i.e. that: Algebraic integers defined for arbitrarily large degrees are uncountable

Nonsense; the set of integer polynomials is countable, so the algebraic integers are a countable union of finite sets. Septentrionalis 14:32, 14 April 2006 (UTC)

Algebraic integers defined for a maximum degree of the polynomial P(x) are not closed under any operation

Depends on the operation. Septentrionalis 14:32, 14 April 2006 (UTC)

Some algebraic integers, such as I = largest real root of (x⁵ - x + 1) cannot be represented without using the polynomial ... Scythe33 02:28, 4 September 2005 (UTC)

I have added Schroeppel's result; since it was shown informally on a mailing list, I won't argue if someone wants to qualify "demonstrated". Septentrionalis 14:32, 14 April 2006 (UTC)

[edit] Noetherian?

Either way, this is overlooking the obvious; but are the algebraic integers Noetherian? Septentrionalis 14:56, 14 April 2006 (UTC)

No. Consider the sequence of principal ideals generated by the elements 2^1/n. Joeldl 10:14, 17 February 2007 (UTC)

[edit] Merger proposal

I propose this article be merged with Integrality, which covers a topic of which this is a special case. Joeldl 10:16, 17 February 2007 (UTC)