Proof. In
order to prove that

is an algebraic extension, we need to show that any element

is
algebraic, i.e., there exists a non-zero
polynomial
![$ p(x)\in K[x]$ $ p(x)\in K[x]$](http://images.planetmath.org:8080/cache/objects/4725/l2h/img6.png)
such that

.
Recall that
is a finite extension of fields, by definition, it means that
is a finite dimensional vector space over
. Let the dimension be
for some

.
Consider the following set of “vectors” in
:
Note that the
cardinality of

is

, one more than the dimension of the vector space. Therefore, the elements of

must be
linearly dependent over

, otherwise the dimension of

would be greater than

. Hence, there exist

, not all zero, such that
Thus, if we define
then
![$ p(X)\in K[X]$ $ p(X)\in K[X]$](http://images.planetmath.org:8080/cache/objects/4725/l2h/img24.png)
and

, as desired.
