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algebraic (Definition)

Let $ B$ be a ring with a subring $ A$. An element $ x \in B$ is algebraic over $ A$ if there exist elements $ a_1, \dots, a_n \in A$, with $ a_n \neq 0$, such that

$\displaystyle a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0. $
An element $ x \in B$ is transcendental over $ A$ if it is not algebraic.

The ring $ B$ is algebraic over $ A$ if every element of $ B$ is algebraic over $ A$.



"algebraic" is owned by djao. [ full author list (2) ]
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See Also: algebraic extension

Also defines:  transcendental
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Cross-references: subring, ring
There are 27 references to this entry.

This is version 4 of algebraic, born on 2002-01-05, modified 2005-03-15.
Object id is 1297, canonical name is Algebraic.
Accessed 8519 times total.

Classification:
AMS MSC: 13B02 (Commutative rings and algebras :: Ring extensions and related topics :: Extension theory)
Pending Errata and Addenda
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