let be a linearly independent subset of i'll prove that choose any maximal ideal of then is a field and thus is a vector space over
the claim is that is a basis for clearly to prove that the elements of are linearly independent, suppose for some
then thus for some this gives us and hence thus for all which proves the claim.
so we've proved that similarly we have that the elements of are linearly independent over but, from linear algebra, we know that the number
of elements of any linearly independent set in a vector space is at most equal to the number of elements of a basis of that vector space. therefore