how do i prove the existence of this norm?

Asked Jun 01 '15 at 02:59

Active Jun 01 '15 at 02:59

Viewed 74 times

I am reading an article that states:

Let $\mathbb{K}$ be a fixed local field. Then there is an integer $q=p^r$, where p is a fixed prime element of $\mathbb{K}$ and r is a positive integer, and a norm $\left | . \right |$ on $\mathbb{K}$ such that for all $x\in \mathbb{K}$ we have $\left | x \right |\geq 0$ and for each $x\in \mathbb{K}$(except 0) we get $\left | x \right |=q^k$ for some integer k. This norm is non-Archimedean, that is $\left | x+y \right |\leq max\left \{ \left | x \right |,\left | y \right | \right \}$ for all $x,y\in \mathbb{K}$ and $\left | x+y \right |=max\left \{ \left | x \right |,\left | y \right | \right \}$ whenever $\left | x \right |\neq \left | y \right |$.