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Galois theory of Salem polynomials

Published online by Cambridge University Press:  28 September 2009

CHRISTOS CHRISTOPOULOS
Affiliation:
Department of Mathematics, Royal Holloway, University of London, Egham Hill, Egham, Surrey, TW20 0EX. e-mail: c.christopoulos@rhul.ac.uk, j.mckee@rhul.ac.uk
JAMES MCKEE
Affiliation:
Department of Mathematics, Royal Holloway, University of London, Egham Hill, Egham, Surrey, TW20 0EX. e-mail: c.christopoulos@rhul.ac.uk, j.mckee@rhul.ac.uk
Corresponding

Abstract

Let f(x) ∈ [x] be a monic irreducible reciprocal polynomial of degree 2d with roots r1, 1/r1, r2, 1/r2, . . ., rd, 1/rd. The corresponding trace polynomial g(x) of degree d is the polynomial whose roots are r1 + 1/r1, . . ., rd + 1/rd. If the Galois groups of f and g are Gf and Gg respectively, then GgGf/N, where N is isomorphic to a subgroup of C2d. In a naive sense, the generic case is GfC2dSd, with NC2d and GgSd. When f(x) has extra structure this may be reflected in the Galois group, and it is not always true even that GfNGg. For example, for cyclotomic polynomials f(x) = Φn(x) it is known that GfNGg if and only if n is divisible either by 4 or by some prime congruent to 3 modulo 4.

In this paper we deal with irreducible reciprocal monic polynomials f(x) ∈ [x] that are ‘close’ to being cyclotomic, in that there is one pair of real positive reciprocal roots and all other roots lie on the unit circle. With the further restriction that f(x) has degree at least 4, this means that f(x) is the minimal polynomial of a Salem number. We show that in this case one always has GfNGg, and moreover that NC2d or C2d−1, with the latter only possible if d is odd.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

[1]Christopoulos, C. On Galois groups of Salem polynomials. PhD. thesis, (University of London 2008).Google Scholar
[2]Curtis, C. W. and Reiner, I.Representation Theory of Finite Groups and Associative Algebras (Wiley, 1988).Google Scholar
[3]Hall, M. Jr., The Theory of Groups (Chelsea, 1976).Google Scholar
[4]Lalande, F.Corps de nombres engendrés par un nombre de Salem. Acta Arith. 88 (1999), 191200.CrossRefGoogle Scholar
[5]Rotman, J. J.An Introduction to the Theory of Groups, Fourth edition. Graduate Texts in Mathematics 148 (Springer, 1995).CrossRefGoogle Scholar
[6]Viana, P. and Veloso, P. M.Galois theory of reciprocal polynomials. Amer. Math. Monthly 109 (2002), No. 5, 466471.CrossRefGoogle Scholar

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