If anyone is interested in the context of this question, I am reading through the proof of Lemma 6.5 in Hartshorne's Algebraic Geometry.

We have fields

such that

. Also, assume

is algebraically closed.

Let

, and let

be the integral closure of the ring

in

. It can be shown that

is a Dedekind domain which is finitely generated as a

-algebra (and I'm fine with that).

Now, suppose

is a discrete valuation ring between

and

such that

. Since

is integrally closed (being a DVR), we must have

.

Here's where I get lost: they say that if

is the (unique) maximal ideal of

, then

is a maximal ideal of

, and that

is

*dominated* by

. What I don't see is (1) how we know that

is a maximal ideal of

(instead of just prime), and (2) why they are saying that

is local. They seem to be claiming this, as the relation of domination is defined only for two local rings.