Do all global fields have infinitely many principal prime ideals?

Question

Let $K$ be a global field, does $K$ have infinitely many principal prime ideals? By a principal prime ideal I mean a finite place $\mathfrak{p}$ for which there exists a $p \in K$ with $\mathfrak{p}$-adic valuation $1$ and $\mathfrak{q}$-adic valuation $0$ for every finite place $\mathfrak{q}$ different from $\mathfrak{p}$.