Let $\mathbb{Q}_p$ be the $p$-adic rational field.
I want to verify that the field extension $\mathbb{Q}_p/ \mathbb{Q}$ is infinite therefore $\dim_{\mathbb{Q}}(\mathbb{Q}_p) = \infty$.
My considerations: Assume the extension is finite. Since $char(\mathbb{Q})=0$ the extension is separable and we can apply the theorem of primitive element and obtain $\mathbb{Q}_p= \mathbb{Q}[T]/f$ for an irreducible $f \in \mathbb{Q}[T]$.
How to obtain the contradiction?
Or is there another way to prove the claim?
Update: Preferably I'm keen curiuos how the agument mentioned by @hunter below using Hensel's lemma (which version) works to solve this problem.