*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, a ^{2},a^{3},...*. 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**.