
> 
> Is there an explicit infinite set of primes, modulo which $X^5 - X - 1$ is irreducible?
> 
> 
> 


Since our polynomial's Galois group over $\mathbb{Q}$ is $S\_5$, Chebotarev's density theorem implies that there are infinitely many such primes, but it is seemingly unclear how to find them.


If the Galois group of a polynomial is abelian, then I think primes modulo which the polynomial is irreducible are just the primes satisfying some congruences.


If we take $X^3-X-1$ instead, then such a set of primes can be constructed using the first answer given in [Galoisian sets of prime numbers](https://mathoverflow.net/questions/11747/galoisian-sets-of-prime-numbers/12382#12382)


As far as I am concerned, once a representation of $S\_5$ is given, the Langlands correspondence should provide some analytic object (e.g modular form) such that the set of primes for which the polynomial is irreducible (the primes which are inert) could be read off this object.


Can recent work on the Langlands correspondence help finding such a set of primes?



> 
> Is there a specific automorphic form which (at least conjecturally) 
>  gives us the required primes?
> 
> 
> 



Explicit Chebotarev and Langlands - irreducibility of X^5-X-1 mod primes
The object (conjecturally) associated to an Artin representation by Langlands is not a classical modular form except in a very limited number of situations (odd two dimensional representations). Let $K$ be the splitting field of some polynomial over $\mathbf{Q}$ with Galois group $S\_5$. If $V$ is any finite dimensional representation of dimension $n$ of $S\_5$, then, associated to 


$$\rho: \mathrm{Gal}(K/\mathbf{Q}) \simeq S\_5 \rightarrow \mathrm{GL}(V)$$


one can write down an Artin $L$-function $L(V,s)$. Then Langlands' extension of the Artin Conjecture predicts that $L(V,s)$ is equal to $L(\pi,s)$, where $\pi$ is an algebraic automorphic representation of $\mathrm{GL}(n)/\mathbf{Q}$ (of some very particular infinity type). For $n \ge 3,$ these objects are not so easy to write down very explicitly. For all practical matters, it is much easier to actually compute the split primes (or primes whose corresponding Frobenius element is any conjugacy class in $S\_5$) directly from the polynomial rather than automorphically. 


In the particular case of the polynomial $X^5 - X - 1$, the Artin conjecture is known for any representation $V$, as follows from a theorem of J. Dwyer (*Real zeros of Artin L-functions corresponding to five-dimensional $S\_5$-representations*, Bull. Lond. Math. Soc. 46 (2014), no. 1, 51–58).


**Response To Comments:**
For computational purposes (say which primes $p < 10000$ are inert) the best method to determine the inert primes is using pari/gp by directly factoring the polynomial modulo $p$. As to what information can be inferred from automorphy, that is hopelessly vague, and depends on what you want to do with it. Unfortunately the margins here are too small for me to explain the entire Langlands programme. The entire point of non-abelian class field theory is that the corresponding Galoisian sets are very complicated. There will be no more "explicit" description of the primes which are inert than "primes such that the Hecke eigenvalues of some appropriate automorphic form $\pi$ have some certain property." 


