Polynomials with roots in for all

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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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Additional Information

**Jack Sonn**

Affiliation:
Department of Mathematics, Technion, 32000 Haifa, Israel

Email:
sonn@math.technion.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-08-09155-7

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