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Here is an interesting problem.

Problem. For what positive integer $p\geq 2$ is the polynomial $f_p(X)=X^{p-1}+X^{p-2}+\cdots +X+1$ irreducible over $\mathbb F_2$?

This problem looks similar to the problem of Mersenne primes. I guess it's not an easy one

Observing that $f_{mn}(X)=f_m(X)f_n(X^m)$, it is clear any such $p$ must be prime, and indeed for $p=2,3,5,11,13$ $f_p(X)$ are irreducible. However, it is not uncommon for a prime $p$ to give a reducible $f_p$: $$\begin{align} f_7(X) &= (X^3+X+1)(X^3+X^2+1) \\ f_{17}(X) &= (X^8 + X^5 +X^4 +X^3 +1)(X^8+X^7+X^6+X^4+X^2+X+1)\\ f_{23}(X) &= (X^{11}+X^9 +X^7+X^6+X^5+X+1)(X^{11}+X^{10}+X^6+X^5+X^4+X^2+1)\\ & \cdots \end{align}$$ and for large $p$ it is actually more common to have a reducible $f_p$.

Some thoughts: if we let $g_p$ be the polynomial in $\mathbb Z[x]$ with the same expression as that of $f_p$, then the Galois group $G\leq S_n$ of $g_p$ over $\mathbb Q$ is generated by a $(p-1)$-cycle. It is not difficult to prove that $f_p(x)$ is separable for all prime $p$. Applying a theorem of Dedekind, if $f_p$ is reducible, then it splits into several polynomials of the same degree. I doubt it's of much use though

So the questions:

  1. How to (partially, in any means) solve the problem? For example, is there infinitely many $p$'s such that $f_p$ is irreducible (or reducible)?
  2. Requesting relevant references
Cyankite
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1 Answers1

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When $p$ is prime, $f_p(x)$ is the $p$th cyclotomic polynomial. Therefore, it is irreducible over $\mathbb F_2$ if and only if $2$ is a primitive root modulo $p$.

According to Artin’s conjecture, there are infinitely many such primes, but as far as I know, it’s not proven yet.

Pranay
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