For which fields $F$ and primes $p$ does the polynomial $x^p-x\in F[x]$ have a “multiple” root

Question

A root $\alpha$ of $f(x)\in F[x]$ is multiple if we can write $(x-\alpha)^m\mid f(x)$ and $m>1$

I found that this is true when $F=\Bbb C$ and $p\geq 3$ because we need the root of our polynomial $f(x)= x^p-x= x(x^{p-1}-1)$ to have an argument $\theta$ such that there exists $m\in\Bbb Z$ : $m\theta\cong 0$ mod $(2\pi)$.

I think there’s a very good chance that I’m wrong...

Answer 1

The root $x=0$ is simple. So when does $x^{p-1}-1$ have multiple roots? Yes, when $F$ has characteristic $q$, and $q\mid(p-1)$. No, otherwise, since the derivative of $x^{p-1}$ is $(p-1)x^{p-2}$ which is nonzero when $x\ne0$.

Answer 2

Over $\mathbb{C}$ the polynomial $x^p - x$ has no multiple roots. In fact, the roots are easy to describe. They are 0 and the $(p-1)$-th roots of unity, which are pairwise distinct.

Lord Shark the Unknown gave a condition for the existence of multiple roots over fields of positive characteristic. Here I would like to make this a little bit more concrete:

The polynomial $x^p - x$ has a multiple root over $\mathbb{F}_2$ for all $p>2$, namely 1. One can even write down a factorization of $x^p - x$. First we have $x^p - x = x(x^{p-1}-1)$, and

$$x^{p-1} - 1 = \prod_{d | p-1} \Phi_d(x)$$

where $\Phi_d(x)$ is the $d$-th cyclotomic polynomial.

Why is 1 a multiple root? The first two cyclotomic polynomials are $\Phi_1(x) = x-1$ and $\Phi_2(x) = x+1$ and $1 \in \mathbb{F}_2$ is a root of both of them, i.e. multiple root of $x^{p-1} - 1$.