Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
I would appreciate it if downvoters could explain the reason for their downvote. – Jesse Madnick Jul 20 '13 at 10:30
up vote 5 down vote accepted

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

share|cite|improve this answer
Thanks, Qiaochu. – Jesse Madnick Oct 8 '12 at 21:31

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.