Polynomials with roots in $\mathbb {Q}_{p}$ for all $p$
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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 $\mathbb {Q}(t)$.References
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Additional Information
- Jack Sonn
- Affiliation: Department of Mathematics, Technion, 32000 Haifa, Israel
- Email: sonn@math.technion.ac.il
- 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
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 1955-1960
- MSC (2000): Primary 11R32, 12F12
- DOI: https://doi.org/10.1090/S0002-9939-08-09155-7
- MathSciNet review: 2383501