Remark

The field K is readily seen to be a vector space over L.

If there is a polynomial h L such that h(a) = 0, then K is a finite-dimensional vector space.

If there is no such polynomial, then K is an infinite-dimensional vector space over L.

For instance, there is no polynomial in Q[X] having as a zero (nontrivial; we give no proof here!), and so K is infinite-dimensional if a = and L = Q.