Let $K$ be a number field, and $p$ a prime in $\mathbb{Z}$. Explain what it means for $p$ to be inert, to split completely, and to be ramified in $K$.

(a) Show that if $[K: \mathbb{Q}]>2$ and $\mathcal{O}_{K}=\mathbb{Z}[\alpha]$ for some $\alpha \in K$, then 2 does not split completely in $K$.

(b) Let $K=\mathbb{Q}(\sqrt{d})$, with $d \neq 0,1$ and $d$ square-free. Determine, in terms of $d$, whether $p=2$ splits completely, is inert, or ramifies in $K$. Hence show that the primes which ramify in $K$ are exactly those which divide $D_{K}$.