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

 

Polynomials with roots in $ \mathbb{Q}_{p}$ for all $ p$


Author: Jack Sonn
Journal: Proc. Amer. Math. Soc. 136 (2008), 1955-1960
MSC (2000): Primary 11R32, 12F12
Published electronically: February 12, 2008
MathSciNet review: 2383501
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Abstract: Let $ f(x)$ be a monic polynomial in $ \mathbb{Z}[x]$ with no rational roots but with roots in $ \mathbb{Q}_{p}$ for all $ p$, or equivalently, with roots mod $ n$ for all $ n$. It is known that $ f(x)$ cannot be irreducible but can be a product of two or more irreducible polynomials, and that if $ f(x)$ is a product of $ m>1$ irreducible polynomials, then its Galois group must be a union of conjugates of $ m$ proper subgroups. We prove that for any $ m>1$, every finite solvable group that is a union of conjugates of $ m$ proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group of such a polynomial, and that the same result (with $ m=2$) holds for all Frobenius groups. It is also observed that every nonsolvable Frobenius group is realizable as the Galois group of a geometric, i.e. regular, extension of $ \mathbb{Q}(t)$.


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Additional Information

Jack Sonn
Affiliation: Department of Mathematics, Technion, 32000 Haifa, Israel
Email: sonn@math.technion.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09155-7
PII: S 0002-9939(08)09155-7
Received by editor(s): January 4, 2007
Received by editor(s) in revised form: March 8, 2007
Published electronically: February 12, 2008
Communicated by: Ken Ono
Article copyright: © Copyright 2008 American Mathematical Society