Proof of a conjecture of S. Chowla (Q5921302)

The following result is proved: for every \(n\ge 2\), the number of elements \(a\) in a finite field \(\mathbb F_q\) of \(q\) elements such that the polynomial \(x^n + x + a\) is irreducible over \(\mathbb F_q\), tends to \(q/n\) as \(q\) tends to \(\infty\), provided that the characteristic of \(\mathbb F_q\) does not divide \(2n(n-1)\). The cases \(n=3,4\) were known before. The proof is a straightforward application of Weil's density theorem and the fact that the Galois group of the polynomial \(f(x) =x^n +x+t\) over \(\mathbb F_q(t)\) is the symmetric group on \(n\) letters. In the proof the author should have shown that the splitting field \(K\) of \(f(x)\) over \(\mathbb F_q(t)\) is a regular extension of \(\mathbb F_q\), to ensure the irreducibility of the covering curve. This could be done easily using the fact \(\G(K/\mathbb F_q(t)) = S_n\) (either using the discriminant of \(f(x)\) or by a direct manipulation of the roots of \(f(x) =0)\). Similarly, in Theorem 2, which is a generalization of the above, one needs the assumption that the splitting field of \(f_t(x)\) over \(\mathbb Q(t)\) is a regular extension of \(\mathbb Q\).