TITLE: $K[[X]]$ is not a finitely generated $K[X]$-module.
QUESTION [15 upvotes]: How can I  prove that $K[[X]]$ is not finitely generated over $K[X]$ as a module, where $K$ is a field.

What I tried: if above is not true then $K[[X]]$ is integral extension over $K[X]$. But I failed to draw any contradiction. 
Help me. Thanks

REPLY [2 votes]: In the following context we assume everything is commutative.

Definition: A Jacobson ring is a ring where every prime ideal is an intersection of some maximal ideals.

Then we have the fact (cf. Eisenbud Commutative Algebra with a View towards Algebraic Geometry Theorem 4.19) that:

Fact (Nullstellensatz): Let $A$ be a Jacobson ring, then any finitely generated $A$-algebra is a Jacobson ring.

Now apply this result to the question. 
First note that a field $K$ is obviously a Jacobson ring, by Nullstellensatz $K[X]$ is a Jacobson ring. Next note that if a local ring has more than one prime ideal, then it cannot be a Jacobson ring. Now $K[[X]]$ is a local ring with prime ideals $(X)$ and $(0)$, hence it cannot be a Jacobson ring. Hence it cannot be a finitely generated $K[X]$-algebra. (A fortiori not finite $K[X]$-module.)