An invariant characterization of pseudo-valuations on a field

Abstract

This note is concerned with pseudo-valuations on a (commutative) field F. A pseudo-valuation on F† is defined as a real-valued function W on F such that(1) W(x) ≥ 0 for all x∈F and W(0) = 0, but W does not vanish identically on F,(2) W(xy) ≤ W(x) W(y) for all x, y ∈ F,(3) W(x−y) ≤ W(x) + W(y) for all x, y ∈ F.W is non-Archimedean, if it satisfies the ultrametric inequality(3′) W(x−y) ≤ max {W(x), W(y)}.