Polynomials with roots in ℚ_{𝕡} for all 𝕡

DOI10.1090/S0002-9939-08-09155-7zbMATH Open1195.12007arXivmath/0612528OpenAlexW1570243798MaRDI QIDQ3506730

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Publication date: 17 June 2008

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Abstract: Let $f(x)$ be a monic polynomial in $dZ[x]$ with no rational roots but with roots in $d{Q}_p$ 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 which 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 $dQ(t)$.

Full work available at URL: https://arxiv.org/abs/math/0612528

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