Let M be a field extension of a field K. An element a in M is called algebraic over K if there exists a nonzero polynomial f(x) with coefficients in K such that f(a)=0. Otherwise a is called transcendental over K.

For example, sqrt(2) is algebraic over Q but pi is not.

We say that M is algebraic over K if every element of M is algebraic over K. Whenever M is finite-dimensional when considered as a vector space over K it is algebraic over K. To see this take a in M and think about the powers of a: 1,a, a2,a3,.... Since these are elements of a finite-dimensional vector space they are linearly dependent. This gives a nonzero polynomial over K with a as a root.

For example,Q(sqrt(2)) is algebraic over Q.