# Is a field extension $L/K$ finite if and only if $L$ is a finitely-generated $K$-algebra?

I recently started learning commutative algebra from Atiyah-MacDonald.

This means that for the next few months, I'll be posting some (mostly silly) questions to check my understanding. (Thank you all in advance for your patience.)

My understanding: Let $L/K$ be a field extension. Then the following are equivalent:

(1) $L$ is a finitely-generated $K$-algebra

(2) $L/K$ is a finite extension

(3) $L$ is a finite $K$-algebra (i.e. finitely generated as a $K$-vector space)

My understanding is that $(2) \iff (3)$ is immediate, as is the implication $(3) \implies (1)$. However, the implication $(1) \implies (2)$ is non-trivial, and is a form of the weak Nullstellensatz.

Is everything I've said correct?

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I would appreciate it if downvoters could explain the reason for their downvote. – Jesse Madnick Jul 20 '13 at 10:30

Yes. ${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$