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algebraic (Definition)

Let $ K$ be an extension field of $ F$ and let $ a\in K$.

If there is a nonzero polynomial $ f\in F[x]$ such that $ f(a)=0$ (in $ K$) we say that $ a$ is algebraic over $ F$.

For example, $ \sqrt{2}\in\mathbb{R}$ is algebraic over $ \mathbb{Q}$ since there is a nonzero polynomial with rational coefficients, namely $ f(x)=x^2-2$, such that $ f(\sqrt{2})=0$.



"algebraic" is owned by drini. [ owner history (1) ]
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See Also: algebraic number, finite extension, proof of transcendental root theorem


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algebraic sum and product (Theorem) by pahio
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Cross-references: coefficients, rational, polynomial, extension field
There are 173 references to this entry.

This is version 3 of algebraic, born on 2001-11-08, modified 2002-05-30.
Object id is 705, canonical name is AlgebraicElement.
Accessed 7117 times total.

Classification:
AMS MSC: 13B05 (Commutative rings and algebras :: Ring extensions and related topics :: Galois theory)
 11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)
 11R32 (Number theory :: Algebraic number theory: global fields :: Galois theory)
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