I just came across the p-adic norm for the first time. I tried to show that it is actually a norm on $Q$ but I was asking myself, whether checking scalability is a bit self referential ?
What I mean by that is, that I am used to show scalability under the absolute value for any scalar from the underlying field (mostly due to my lack of experience this was in the context where there is a clear intuitive interpretation for the absolute value, e.g. on $\mathbb{R} \quad or \quad \mathbb{C}$, but with the p-adic the scalability criterion is only satisfied w.r.t the p-adic itself which seems self referential to me. Can somebody help clarify this notion ?