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Introduction

In: Diophantine Equations and Power Integral Bases

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  • István Gaál

    (University of Debrecen, Institute of Mathematics and Informatics)

Abstract

Let K be an algebraic number field of degree n with ring of integers ℤ K . The ring ℤ K is called monogenic if it is a simple ring extension ℤ[α] of ℤ. In this case 1, α,...,αn-1 is an integral basis of K, called a power integral basis. Our main task is to develop algorithms for determining all generators α of power integral bases. As we shall see, this algorithmic problem is satisfactorily solved for lower degree number fields (especially for cubic and quartic fields) and there are efficient methods for certain classes of higher degree fields. Our algorithms enable us in many cases to describe all power integral bases also in infinite parametric families of certain number fields.

Suggested Citation

  • István Gaál, 2002. "Introduction," Springer Books, in: Diophantine Equations and Power Integral Bases, chapter 1, pages 1-5, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-0085-7_1
    DOI: 10.1007/978-1-4612-0085-7_1
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