Valuation of a complete field is $\mathbb{Z}$?

Question

I was studying the book Local Class field theory by Iwasawa. There it's written if $(k,v)$ is a complete field (by what I mean is $k$ is complete under $v$ topology given by valuation $v$ on $k$) then $v(k^*)=\mathbb{Z}$. Can anyone please explain why, I can't figure it out

Here's what eating me..

Answer 1

That's not true in general.

If $k=\Bbb Q_p$ and $v$ is the $p$-adic valuation, then $v(\Bbb Q_p^\times)=\Bbb Z$. But if $k=\hat{\Bbb C}_p$ the $p$-adic valuation is not discrete.