Unramified extension of $\mathbb{Q}_p$

Question

Let $K$ be an unramified (finite) extension from $\mathbb{Q}_p$. Is is equivalent to saying $\mathbb Q_p\to K$ is an unramified ring homomorphism? Or it is equivalent to saying $\mathbb Z_p\to \mathcal O_K$ is unramified ring homomorphism?

Answer 1

When $k'/k$ is a finite separable extension of fields, the corresponding homomorphism $k \to k'$ is always unramified.

Suppose that $K/\mathbb{Q}_p$ is unramified. Then this is the case if and only if the maximal ideal $\mathfrak{m}$ of $\mathcal{O}_K$ is equal to $p\mathcal{O}_K$, (the extension of residue fields is always separable) so $\mathbb{Z}_p \to \mathcal{O}_K$ is an unramified local homomorphism if and only if $K/\mathbb{Q}_p$ is unramified.

Since both rings are complete with respect to their maximal ideals being an unramified local homomorphism is equivalent to being an unramified ring homomorphism see StacksProject Lemma 41.3.4.