TITLE: $f \in \mathbb C[x]$ monic polynomial of prime degree which is not co-prime with any of its derivative polynomials, then $f$ has only one root?
QUESTION [9 upvotes]: Let $f \in \mathbb C[x]$ be a monic polynomial of prime degree $p$ such that $f$ is not co-prime with any $f^{(k)}$ ($k$-th derivative) for $1\le k<p$. Then is it true that $\exists a\in \mathbb C$ such that $f(x)=(x-a)^p$ ?

REPLY [3 votes]: I think that this is a solved particular case of an open problem, called the Casas-Alvero conjecture (from the Spanish mathematician Eduardo Casas-Alvero, proposed in 2001)!

Casas-Alvero conjecture. Let $f$ be a polynomial of degree $d$ defined over a field $K$ of characteristic zero. If $f$ has a factor in common with each of its derivatives $f^{(i)}$, $i = 1, \dots, d − 1$ then the conjecture predicts that $f$ must be a power of a linear factor.

Notice that the common factor of $f$ and $f^{(i)}$ (or the common root if $K$ is algebraically closed) may depend on $i$.
But hopefully, the case where $d=p^k$ is a prime power (in particular if $f$ has prime degree) has been solved by Hans-Christian Graf von Bothmer, Oliver Labs, Josef Schicho, Christiaan van de Woestijne
in The Casas-Alvero conjecture for infinitely many degrees, Journal of Algebra 316(1), pp. 224-230 (2007).