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