Rank-one valuation
= Rank-one valuation
{title2=$v:K^\times\to\Gamma\subseteq\mathbb R$}
A nontrivial additive <valuation> has rank one when its ordered value group is Archimedean and therefore embeds order-preservingly in $\mathbb R$. Exponentiating a real-valued representative gives a <Non-Archimedean absolute value>; changing the embedding rescales the valuation positively. Higher-rank ordered value groups cannot be represented by real-valued <absolute values on a field>.