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.