Polynomials with roots in for all ![$ p$](/proc/2008-136-06/S0002-9939-08-09155-7/gif-title0/img2.gif) |
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Authors: | Jack Sonn |
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Institution: | Department of Mathematics, Technion, 32000 Haifa, Israel |
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Abstract: | Let be a monic polynomial in with no rational roots but with roots in for all , or equivalently, with roots mod for all . It is known that cannot be irreducible but can be a product of two or more irreducible polynomials, and that if is a product of irreducible polynomials, then its Galois group must be a union of conjugates of proper subgroups. We prove that for any , every finite solvable group that is a union of conjugates of proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group of such a polynomial, and that the same result (with ) 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 . |
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Keywords: | |
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