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
- $1 iff 2$. 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 $mathfrak m = langle x_1, x_2, ldots, x_nrangle$ 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?