Mihăilescu’s Ideal

Yuri F. Bilu, Yann Bugeaud and Maurice Mignotte
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Yuri F. Bilu: University of Bordeaux and CNRS, Institute of Mathematics of Bordeaux
Yann Bugeaud: University of Strasbourg and CNRS, IRMA, Mathematical Institute
Maurice Mignotte: University of Strasbourg and CNRS, IRMA, Mathematical Institute

Chapter Chapter 8 in The Problem of Catalan, 2014, pp 97-115 from Springer

Abstract: Abstract Let (x, y, p, q) be a solution of Catalan’s equation (with x, y nonzero integers and p, q odd primes) and G the Galois group of the cyclotomic field K = ℚ ( ζ p ) $$K = \mathbb{Q}(\zeta _{p})$$ . From the previous chapters we know about the important role of the elements Θ ∈ ℤ [ G ] $$\varTheta \in \mathbb{Z}[G]$$ such that (x −ζ) Θ (or, equivalently, (1 −ζ x) Θ ) is a qth power in K. Elements Θ with this property form an ideal of the group ring ℤ [ G ] $$\mathbb{Z}[G]$$ , called Mihăilescu’s ideal. It is convenient to study Mihăilescu’s ideal on its own, without any reference to Catalan’s equation. In this chapter we establish some properties of Mihăilescu’s ideal and apply them to Catalan’s equation.

Keywords: Galois Group; Nonzero Integer; Tijdeman; Hanrot; Complex Algebraic Numbers (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-10094-4_8

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DOI: 10.1007/978-3-319-10094-4_8

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