# algebraic

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 .

Title algebraic Algebraic 2013-11-05 18:32:06 2013-11-05 18:32:06 drini (3) pahio (2872) 8 drini (2872) Definition msc 13B05 msc 11R04 msc 11R32 AlgebraicNumber FiniteExtension ProofOfTranscendentalRootTheorem