How do I prove the existence of this norm?

[zarei175]

I am reading an article[1] that states:

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

 

[mathwonk]

I recommend say the first 8-10 pages of these notes of Pete Clark:

http://math.uga.edu/~pete/8410FULL.pdf

 

[zarei175]

Thank you mathwonk.. I read these pages, but my question is different.