I am reading an article[1] that states:(adsbygoogle = window.adsbygoogle || []).push({});

Let k be a fixed local field. Then there is an integer q=p^{r}, where p is a fixed prime element of k and r is a positive integer, and a norm |.| on k such that for all x∈k we have |x|≥0 and for each x∈k\{0} we get |x|=q^{m}for some integer m. This norm is non-Archimedean, that is |x+y|≤max{|x|,|y|} for all x,y∈k and |x+y|=max{|x|,|y|} whenever |x|≠|y|.

how do i prove the existence of this norm?

and

how to prove that the norm is non-Archimedean?

[1]: http://docdro.id/11a73

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# How do I prove the existence of this norm?

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