# p adic number theory – Reference request for characterisation of discrete valuations

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 $$mathfrak m = langle x_1, x_2, ldots, x_nrangle$$ 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?