A <discrete valuation> on a field $K$ is a surjective group homomorphism
$$
v:K^\times\longrightarrow\mathbb Z
$$
satisfying
$$
v(x+y)\geq\min\{v(x),v(y)\}
$$
whenever $x+y\ne0$. Its <discrete valuation ring> is
$$
\boxed{R_v=\{0\}\cup\{x\in K^\times:v(x)\geq0\}}.
$$

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.

Solution

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

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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-101/5/i/solution