Reference for detailed proof of non equivalent norms over dimensionally finite field,$\mathbb{Q}(\sqrt{2})$ example

Question

The following post gives the example on $\mathbb{Q}(\sqrt{2})$

https://mathoverflow.net/questions/265192/non-equivalent-norms-on-finite-dimensional-vector-spaces-over-a-non-complete-fie

I'd like to know if someone has a book reference for a detailed proof about the non equivalence of these norms and that would give maybe some more details.