A journey through Galois groups, irreducible polynomials and diophatine equations

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Authors
Filaseta, M.
Luca, F.
Stănică, P.
Underwood, R.G.
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Date of Issue
2017
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Abstract
Computing the Galois group of the splitting field of a given polynomial with integer coefficients over the rationals is a classical problem in modern algebra. A theorem of Van der Waerden [Wae] asserts that almost all (monic) polynomials in Z[x] have associated Galois group Sn, the symmetric group on n letters. Thus, cases where the associated Galois group is different from Sn are rare. Nevertheless, examples of polynomials where the associated Galois group is not Sn are well-known. For example, the Galois group of the splitting field of the polynomial xp − 1, p 3 prime, is cyclic of order p − 1. For the polynomial xp − 2, p > 3, the Galois group is the subgroup of Sp generated by a cycle of length p and a cycle of length p − 1. One interest in this paper is to find other collections of polynomials with integer or rational coefficients whose Galois groups are isomorphic to these groups.
Type
Article
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Department
Applied Mathematics
Organization
Naval Postgraduate School (U.S.)
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NPS Report Number
Sponsors
Natonal Science Foundation (Filaseta)
Funder
Grants SEP-CONACyT 37259 E and 37260E (Luca)
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16 p.
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This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.
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