Exercise 22, § 8

a) If there exists a discrete valuation on a field $K$, prove that the algebraic closure of $K$ is of infinite degree over $K$.

b) Let $K$ be a finite type extension of a field $K_0$. Prove that if $K$ is not algebraic over $K_0$, there exists a discrete valuation $\varphi$ on $K$ such that $\varphi(x) = 0$ in $K_0$.