Statement
Let be a field extension.
If is finitely generated as a -algebra, then is finite as an extension.
Proof
Uncountable version
Let be an uncountable field.
Say be such that is a finitely generated -algebra.
Then for some and ideal
Then
and is finitely generated extension.
This still does not mean that is finite (that the degree is finite), to prove this it suffices to show that it is algebraic extension since for finitely generated extensions, algebraic implies finite.
Let .
Not has a countable basis since it is generated by monomials which are countable.
Consider the set
Since is uncountable and has countable dimension (over ) then there exists a linear relationship
We can take this expression and combine to make a “common denominator” which will look something like
where and is the function that comes from clearing denomiators.
This gives that is a root of this function, so it is algebraic and thus finite.