I have reading the book "$p$-adic Numbers An Introduction" of Gouvêa and I didn't understand a proof. In particular, Theorem 3.2.13 states that
The field $\mathbb{Q}_{p}$ is unique up to unique isomorphism preserving the absolute value.
If we have another completion $K$, then we can think of the inclusion $\mathbb{Q}\hookrightarrow K$ as a map defined on a subset of $\mathbb{Q}_{p}$. We know that is continous and the extension is unique (because $\mathbb{Q}$ is dense). The book suggests to perform the same process and then we can consider the inclusion $\mathbb{Q}\rightarrow\mathbb{Q}_{p}$ with $\mathbb{Q}\subset K$. Again, $\mathbb{Q}$ is dense and we can uniquely extend. Now the book says that the restriction is the identity of $\mathbb{Q}$ (and this is clear), but I don't see why this is enough to say that we found the inverse. Can someone help me? Thanks before!
Consider the composition $\phi \colon \mathbb{Q}_p \to K \to \mathbb{Q}_p$ of your constructed homomorphisms. By construction of the morphisms, $\phi$ is the identity when restricted to $\mathbb{Q}$. Now $\operatorname{id}, \phi \colon \mathbb{Q}_p \to \mathbb{Q}_p$ are two extensions of the identity on $\mathbb{Q}$. As you have remarked, $\mathbb{Q}$ is dense in $\mathbb{Q}_p$, so extensions are unique. This shows $\phi = \operatorname{id}$.
The same argument shows that $K \to \mathbb{Q}_p \to K$ is also the identity.