What I am asking for might be impossible, anyway let me try. Long time ago (between 2015 and 2019?) I remember reading a book, or maybe an article, related to $p$-adic numbers in which following proposition was given:
Let $v$ be a valuation on field $K$. The following statements are equivalent:
- $v$ is discrete, that is $v(K^\times) \cong \mathbb Z$,
- ideal $\mathfrak m = (\pi)$ is principal
- valuation ring $\mathcal O$ is a PID
- valuation ring $\mathcal O$ is noetherian
Proof:
- $1 \iff 2$. Generators of the $\mathfrak m$ ideal are elements $\pi \in K$ of minimal positive valuation
- $2 \implies 3$. Follows from $2 \implies 1$ because in every ideal there is an element with minimal valuation.
- $3 \implies 4$. If every ideal is principal, then it's in particular finitely generated, and so the ring is noetherian
- $4 \implies 2$. Let $\mathfrak m = \langle x_1, x_2, \ldots, x_n\rangle$ be an ideal and wlog assume that $v(x_1) \le \ldots \le v(x_n)$. Then $v(x_i/x_1) \ge 0$ and so $x_i/x_1 \in \mathcal O$, so $\mathfrak m = \langle x_1 \rangle$.
What was its title/author?