# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. · Zbl 0175.03601  Lior Bary-Soroker and Moshe Jarden, PAC fields over finitely generated fields, to appear in Math. Z. · Zbl 1146.12002  Claude Chevalley, Introduction to the Theory of Algebraic Functions of One Variable, Mathematical Surveys, No. VI, American Mathematical Society, New York, N. Y., 1951. · Zbl 0045.32301  John D. Dixon and Brian Mortimer, Permutation groups, Graduate Texts in Mathematics, vol. 163, Springer-Verlag, New York, 1996. · Zbl 0951.20001  Michael Fried, Dan Haran, and Moshe Jarden, Galois stratification over Frobenius fields, Adv. in Math. 51 (1984), no. 1, 1 – 35. · Zbl 0554.12016  Michael D. Fried and Moshe Jarden, Field arithmetic, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 11, Springer-Verlag, Berlin, 2005. · Zbl 1055.12003  Michael D. Fried and Helmut Völklein, The embedding problem over a Hilbertian PAC-field, Ann. of Math. (2) 135 (1992), no. 3, 469 – 481. · Zbl 0765.12002  Moshe Jarden and Aharon Razon, Pseudo algebraically closed fields over rings, Israel J. Math. 86 (1994), no. 1-3, 25 – 59. · Zbl 0802.12007  Heinrich Kornblum, Über die Primfunktionen in einer arithmetischen Progression, Mathematische Zeitschrift 5 (1919), 100-111. · JFM 47.0154.02  Gunter Malle and B. Heinrich Matzat, Inverse Galois theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999. · Zbl 0940.12001  Hugh L. Montgomery and Robert C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. · Zbl 1142.11001  Michael Rosen, Number theory in function fields, Graduate Texts in Mathematics, vol. 210, Springer-Verlag, New York, 2002. · Zbl 1043.11079  Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. · Zbl 0423.12016  Jean-Pierre Serre, Topics in Galois theory, Research Notes in Mathematics, vol. 1, Jones and Bartlett Publishers, Boston, MA, 1992. Lecture notes prepared by Henri Damon [Henri Darmon]; With a foreword by Darmon and the author. · Zbl 0746.12001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.