Cardano, Casus Irreducibilis, and Finite Fields

Publisher： Mathematical Association of America

E-ISSN： 1930-0980|87|5|377-380

ISSN： 1930-0980

Source： Mathematics Magazine, Vol.87, Iss.5, 2014-12, pp. : 377-380

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Abstract

For centuries, mathematicians were puzzled by the fact that Cardano's formula requires an excursion into the complex numbers to express in radicals the roots of an irreducible cubic polynomial with rational coefficients having only real roots. In this paper, we consider an analogous case for irreducible cubics over finite fields. In this simpler setting, we explore whether the roots of an irreducible cubic polynomial are expressible in terms of irreducible radicals present in its splitting field. The key theorem gives a simple divisibility condition that answers this question completely. Thus there is a situation that mirrors the historic “casus irreducibilis,” in that both cases require the presence of elements not in the splitting field in order to express the roots of an irreducible cubic in terms of irreducible radicals.