The multiplicative group of a local field $K$ with valuation ring $\mathcal{O}$ and residue class field of $\overline{K}$ of degree $q=p^f$ splits as
$K=\langle \pi\rangle\times \mu_{q-1}\times U^{(1)}$,
where $\mu_{q-1}$ are the $q-1$ roots of unity in $\mathcal{O}$, $\pi$ a uniformizer and $U^{(1)}$ the principal units.
I was wondering what we can say about the number of roots of unity in $K$? This boils down to finding the possible roots of unity contained in $U^{(1)}$. If $1+x\in U^{(1)}$, then
$(1+x)^n=\sum_{k=0}^n {n\choose k}x^n$,
so showing that this equals $1$ would boil down to proving that
${n\choose 1}x+{n\choose 2}x^2+\ldots+{n\choose n}x^n$
is zero. Is this ever possible? I've tried proving it, but the problem seems to be if $\pi\mid {n\choose k}$ for a lot of different $k$, so we can't necessarily show that one of the above terms would have a larger absolute value than the rest, which would imply that this is impossible.
Can anyone elaborate on the number of roots of unity?