First we consider the case , i.e. we must show ,
whereby it is sufficient to find .
By definition, iff .
We will try to find so that , but ,
so that is the appropriate .
To this end, let .
By ^P1,
we have for some nonzero prime ideals,
where we are free to assume that is minimal.
Since it follows by ^D2.
say .
But since , is maximal, so .
If , then again by maximality ,
whence cannot be equal to or else is a unit and which is not prime.
Now consider ,
whence by minimality of .
Hence there exists such that .
By construction , and it follows that ,
proving the case .
More generally, use the Noetherian nature of to write .
Suppose towards contradiction .
Then for every , we may write
for some .
Let , so that
whence .
But is a monic polynomial in with coëfficients in , whence is integral over .
But is integrally closed, hence , contradicting the above special case.
For invertibility, note for all ,
and , so .
Since but is an ideal, and is maximal, it follows .