TITLE: Is the derivative of $x^n + x^{n-1} + \dots + x + 1$ irreducible?
QUESTION [26 upvotes]: I am working on some combinatorics problems. One of my problems leads to the following question:
Is it true that the derivative of $x^n + x^{n-1} + \dots + x + 1,$ namely $nx^{n-1} + (n-1)x^{n-2} + \dots + 2x + 1$, is irreducible in $\mathbb{Z}[X]$?
I believe it is true, and I have test by computers that it is true for $n \leq 100.$

REPLY [10 votes]: (Turning the comments into a community wiki answer.)
This problem is discussed in Classes of polynomials having only one non-cyclotomic irreducible factor, by Borisov, Filaseta, Lam, and Trifonov (Acta Arithmetica 90 (1999), 121–153).  They prove irreducibility in many special cases but the problem remains open (or at least, none of the papers that Google Scholar lists as citing this paper solves the problem).