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, $\sqrt{2}\in ℝ$ 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.

Title	algebraic
Canonical name	Algebraic
Date of creation	2013-11-05 18:32:06
Last modified on	2013-11-05 18:32:06
Owner	drini (3)
Last modified by	pahio (2872)
Numerical id	8
Author	drini (2872)
Entry type	Definition
Classification	msc 13B05
Classification	msc 11R04
Classification	msc 11R32
Related topic	AlgebraicNumber
Related topic	FiniteExtension
Related topic	ProofOfTranscendentalRootTheorem