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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Dirichlet's theorem for polynomial rings


Author: Lior Bary-Soroker
Journal: Proc. Amer. Math. Soc. 137 (2009), 73-83
MSC (2000): Primary 12E30, 12E25
Published electronically: August 13, 2008
MathSciNet review: 2439427
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Abstract: We prove the following form of Dirichlet's theorem for polynomial rings in one indeterminate over a pseudo algebraically closed field $ F$. For all relatively prime polynomials $ a(X), b(X)\in F[X]$ and for every sufficiently large integer $ n$ there exist infinitely many polynomials $ c(X)\in F[X]$ such that $ a(X) + b(X)c(X)$ is irreducible of degree $ n$, provided that $ F$ has a separable extension of degree $ n$.


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Additional Information

Lior Bary-Soroker
Affiliation: School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978 Israel
Address at time of publication: Department of Mathematics, The Hebrew University, Givat Ram, Jerusalem 91904, Israel
Email: barylior@post.tau.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09474-4
PII: S 0002-9939(08)09474-4
Keywords: Dirichlet's theorem, arithmetic progression, field arithmetics, Hilbert's irreducibility theorem, PAC field
Received by editor(s): January 29, 2007
Received by editor(s) in revised form: July 23, 2007, September 11, 2007, and January 2, 2008
Published electronically: August 13, 2008
Communicated by: Ted Chinburg
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.