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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 $$y$$.

MSC:
 1.2e+31 Field arithmetic 1.2e+26 Hilbertian fields; Hilbert’s irreducibility theorem
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References:
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