# how can I prove this statement in local field theory?

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  such that for each $x\in\mathbb{K}, x\neq0$, we have $|x|=q^k$ for some integer $k$. [/edit]

• What is $q{}{}$? – Wojowu Dec 24 '15 at 13:37
• Do you mean this: “If $\Bbb K$ is a local field of characteristic zero, and $\pi$ is a prime element of $\Bbb K$, then there is $r\in\Bbb Z$ such that $\pi^r\in\Bbb Z.\quad$” ? – Lubin Dec 24 '15 at 14:23
• I am reading this article docdro.id/8Vg64lu. The first two lines of page 295 is devoted to the Subject. – Ehsan zarei Dec 24 '15 at 15:33
• The presentation of $p$-adic facts in the paragraph you’re reading is very unsatisfactory. In particular, $p$ is an prime integer that is determined from the field $\Bbb K$—it’s the characteristic of the residue field— and $q$ is a specific power of that. Let me say that if you’re hoping to learn something about $p$-adic numbers, that article is not the place. They have written for people who already know about local fields, and are reminding their readers of the facts. – Lubin Dec 24 '15 at 16:00
• I added a sentence introducing $q$ from your source. Without it the question was IMHO non-sensical. I think that your title also needs work given that this is really about the theory of local fields. – Jyrki Lahtonen Dec 26 '15 at 23:41