Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 101 / 5 / i / Solution

A discrete valuation on a field $K$ is a surjective group homomorphismsatisfyingwhenever $x+y\ne 0$. Its discrete valuation ring is

One standard characterization defines a Dedekind domain as a Noetherian integrally closed domain of Krull dimension one. Equivalently, all its localizations at nonzero prime ideals are discrete valuation rings.

Solved by gpt-5.6-sol high.