$p$-roots of unity

Question

Let $K$ be a field of characteristic zero complete with respect to a non archimedian absolute value with a residue field of characteristic $p>0$. I would like to show that if $K$ contains the $p$-roots of unity then it also contain $p-1$ distinct non zero roots of the equation $X^p=-pX$. Thanks for the help !

Answer 1

This follows from Krasner's lemma.

The 'right-hand (larger) side' of Krasner's lemma:

If $\alpha$ of $\beta$ are distinct roots of $X^{p-1}=-p$ then $\beta = \zeta_{p-1}\alpha$, where $\zeta_{p-1}$ is a $(p-1)$th root of unity (which, of course, belongs to ${\mathbb Q}_p$), and not equal to one. Therefore $$|\alpha-\beta| = |\alpha|\cdot|1-\zeta_{p-1}|.$$ Now, the second norm on the right is equal to one [$ x^{p-1} -1 $ has distinct non-zero roots $\pmod p$], therefore $$|\alpha-\beta| = |p|^{1/(p-1)}.$$

The 'left-hand (smaller) side' of Krasner's lemma:

Now, set $\pi = \zeta_p -1$, with $\zeta_p$ a primitive $p$th root unity, and take the (shifted cyclotomic) polynomial $$f(x) = {(x+1)^p-1 \over x}. $$ Then $f(x) \equiv x^{p-1} \pmod p$, $f(\pi)=0$, and $f(0 )=p$.

Therefore $-\pi^{p-1} = p\pi (\cdots) + p.$ Hence $$\alpha^{p-1} - \pi^{p-1} = -p + p + p\pi (\cdots),\tag{*}$$ and $$|\alpha^{p-1} - \pi^{p-1}|\le |p\pi|.$$ Now the left of $({}^*)$ can be completely factored, with factors of the form $\alpha -\zeta_{p-1}\pi$. At least one of the factors has norm (strictly) less than $ |p|^{1\over (p-1)}$.

Hence Krasner's lemma applies.