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2008 On the Construction of Families of Cyclic Polynomials Whose Roots Are Units
F. Thaine
Experiment. Math. 17(3): 315-331 (2008).

Abstract

For several values of $m$, we show ways to construct some families of cyclic monic polynomials of degree $m$ with integer coefficients and constant terms $\pm 1$, and to express their roots in terms of Gaussian periods. We give several examples illustrating those techniques. With the aim to find other methods to construct such families of polynomials, we consider the question whether one of them can be obtained by means of rational transformations of a given ordinary family $F(t,X)\in\Z[t][X]$ (at the parameter $t$) of cyclic monic polynomials of degree $m$ (such families $F$ are easy to find). We show a method, due to René Schoof, that allows us to answer at least the simpler question whether a family $G(t,X)\in\Q[t][X]$ of cyclic monic polynomials of degree $m$ prime with constant term in $\Q^\times$ (independent of $t$ can be obtained from $F$} by means of rational transformations.

Citation

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F. Thaine. "On the Construction of Families of Cyclic Polynomials Whose Roots Are Units." Experiment. Math. 17 (3) 315 - 331, 2008.

Information

Published: 2008
First available in Project Euclid: 19 November 2008

zbMATH: 1219.11159
MathSciNet: MR2455703

Subjects:
Primary: 11R09 , 11R18 , 11T22
Secondary: 11R20 , 12F10

Keywords: Cyclic polynomials , Gaussian periods , units

Rights: Copyright © 2008 A K Peters, Ltd.

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Vol.17 • No. 3 • 2008
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