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Dirichlet’s theorem for polynomial rings. (English) Zbl 1259.12002
Artin proved the following strengthening of Dirichlet’s theorem for polynomials over finite fields. For every two polynomials \(a(x), b(x)\) over a finite field \(F\) and every sufficiently large integer \(n\), there is a degree \(n\) polynomial \(c(x)\) such that \(a(x) + b(x) c(x)\) is irreducible. The main theorem states that the same result holds over pseudo-algebraically closed (PAC) fields, for integers \(n\) such that \(F\) has an extension of degree \(n\). A field \(F\) is called PAC if every nonempty absolutely irreducible variety defined over \(F\) has an \(F\)-rational point. These fields play an important role in field arithmetic.
The proof is based on a construction of polynomials \(c(x)\) over any infinite field \(F\) such that \(a (x) + b(x) c (x) y\) is an irreducible polynomial over \(\tilde{F}(y)\) with \(\text{Gal} (f,\tilde{F}(y))= S_n\), where \(\tilde{F}\) is a separable closure of \(F\). This construction is combined with a “weak” Hilbert irreducibility theorem over PAC fields to obtain the desired result by specializing \(y\).

12E30 Field arithmetic
12E25 Hilbertian fields; Hilbert’s irreducibility theorem
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