13B05-Algebraic.tex

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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 \emph{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 {\it field extension} $K/F$ is {\it algebraic}.
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