I am studying an article by Koji Sekiguchi and am stuck at a corollary that states:
$Zar(K) =\{K\} $ if and only if $K$ is an algebraic extension over a finite field. Where $Zar(K)$ denotes the set of all valuation rings with quotient field $K$
This result follows a lemma that states:
Let $K$ be a field and $A$ a subring of $K$. The next three conditions are equivalent:
-$Zar(K\mid A)$={$K$} $\ \ \ $ (($Zar(K\mid A)$ is simply the set of valuation rings containing the ring $A$))
-$K$ is an integral extension over $A$.
-$A$ is a field and $K$ is an algebraic extension over $A$.
I am still struggling with how is it sufficient for $K$ to be an algebraic extension over just one finite field, in order for it to contain no valuation rings. Here's a citation for the article:
Sekiguchi, Koji, Prüfer domain and affine scheme, Tokyo J. Math. 13, No. 2, 259-275 (1990). ZBL0726.14001.