Non-equivalent norms on finite dimensional vector spaces over a non-complete field

Question

It is widely known that on a finite-dimensional vector space over a complete field, every norm is equivalent. However, I'm trying (and failing) to find a counterexample over a field which is not complete.

My first try was to treat $\mathbb{Q}$ as a vector space over itself and find two non-equivalent norms. As for Ostrowski's theorem we know that, for example, the absolute value and the 2-adic norm are not equivalent. However, I noticed that the $p$-adic norm, though being a norm on the field $\mathbb{Q}$, it is not a norm on the vector space $(\mathbb{Q},|\cdot |)$.

I have tried to find such norms in $\mathbb{Q}^2$ and other (not complete) fields to no avail. I'm content with any example, but it would be great if it's accompanied by the proof of their non-equivalence.

Answer 1

The question has been answered in the comment section, which should not be used for giving answers. Let me summarize the comments then.

1. https://mathoverflow.net/questions/265192 gives examples of norms on $\mathbb{Q}(\sqrt{2})$ over $\mathbb{Q}$ which are not equivalent.
2. If $f \in \mathbb{Q}[x]$ is an irreducible polynomial which has a root $\alpha \in \mathbb{C} \setminus \mathbb{R}$ and a root $\beta \in \mathbb{R}$, then $\mathbb{Q}[x]/\langle f \rangle$ can be given two non-equivalent norms.