how do i prove the existence of this norm? (1 Viewer)

[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

 

[Silly Sausage]

You've came to the wrong place.

 

[zarei175]

You've came to the wrong place.

Sorry. Symbols were wrong. I edit symbols.

 

[Silly Sausage]

Generally questions of this of complexity won't be answered. Have a look on physics forums or some other math forum for these types of questions, this website is typically used for high school students. https://www.physicsforums.com/forums/calculus-and-beyond-homework.156/