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, 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**.