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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:
 [1] 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 [2] Lior Bary-Soroker and Moshe Jarden, PAC fields over finitely generated fields, to appear in Math. Z. · Zbl 1146.12002 [3] 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 [4] John D. Dixon and Brian Mortimer, Permutation groups, Graduate Texts in Mathematics, vol. 163, Springer-Verlag, New York, 1996. · Zbl 0951.20001 [5] Michael Fried, Dan Haran, and Moshe Jarden, Galois stratification over Frobenius fields, Adv. in Math. 51 (1984), no. 1, 1 – 35. · Zbl 0554.12016 [6] 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 [7] 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 [8] Moshe Jarden and Aharon Razon, Pseudo algebraically closed fields over rings, Israel J. Math. 86 (1994), no. 1-3, 25 – 59. · Zbl 0802.12007 [9] Heinrich Kornblum, Über die Primfunktionen in einer arithmetischen Progression, Mathematische Zeitschrift 5 (1919), 100-111. · JFM 47.0154.02 [10] Gunter Malle and B. Heinrich Matzat, Inverse Galois theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999. · Zbl 0940.12001 [11] 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 [12] Michael Rosen, Number theory in function fields, Graduate Texts in Mathematics, vol. 210, Springer-Verlag, New York, 2002. · Zbl 1043.11079 [13] 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 [14] 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
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