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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 $f(\sqrt{2})=0$.

If all elements of $K$ are algebraic over $F$, one says that the field extension $K/F$ is algebraic.

Related:

AlgebraicNumber, FiniteExtension, ProofOfTranscendentalRootTheorem

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

13B05*no label found*11R04

*no label found*11R32

*no label found*

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