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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

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?

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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."

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  • $\begingroup$ Great! But how is "it is much easier to actually compute the split primes directly from the polynomial" done in practice (note also that I was asking about inert primes). Also, what arithmetic information we infer from the existence of the automorphic form? $\endgroup$ – Pablo Aug 15 '15 at 13:53
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In order to use the known cases of Langlands the starting point is Serre's conjecture, now a theorem of Khare and Wintenberger. Serre's conjecture states that any odd, continuous, absolutely irreducible, dimension two over a finite field, representation of $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ is modular.

So in your problem we need to find a field and a representation of dimension two of $S_5$. Unfortunately there is an obvious projective representation, but no obvious representation. Results of Tate that were cited by Bosman in his PhD thesis provide conditions under which lifting is possible, namely that we can take unramified liftings at all but finitely many primes. However, it isn't clear to me what that lifting would look like in this case.

Once we find the representation (ideally with some explicit formula) we have to ask what sort of automorphic object appears on the other side. In this case it will be a modular form, which we know how to write down and calculate, once you know the weight, level, and enough coefficients to give it uniquely.

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    $\begingroup$ Let's say we have that modular form and we completely understand it. How do we extract such a set of primes as the OP asked for from it? $\endgroup$ – davidlowryduda Aug 10 '15 at 20:36
  • $\begingroup$ A congruence class is determined by its traces in all representations. So we may need several modular forms, but once we have them we can say "a prime $p$ has Frob in this congruence class if and only if the coefficients have certain values". You might not be satisfied with this answer, and in some ways I'm not either, but this is analogous to what class field theory gives us. $\endgroup$ – Watson Ladd Aug 11 '15 at 2:49
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    $\begingroup$ $S_{5}$ has no faithful representation of degree $2$ over any field, while $A_{5}$ has no faithful representation of degree $2$ over any field of characteristic different from $2$ (we do though have $A_{5} \cong {\rm SL}(2,4)$, for example). $\endgroup$ – Geoff Robinson Aug 11 '15 at 21:28

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