I can understand the (little) reasoning to the point of supposing that $x$ belongs in the discrete valuation ring. Assuming that $\mathfrak{p}$ is the (to be proven) maximal ideal, I don't understand what's the reasoning there.
Could anyone point out what I'm missing here?
Thanks in advance!
Edit: I should mention two things.
- The example can be found in "Introduction to Algebraic and Abelian Functions" by S. Lang.
- I'm open to different proofs of the given statement.