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?