On Index and Monogenity of Certain Number Fields Defined by Trinomials
Lhoussain El Fadil
Abstract
ABSTRACT Let K be a number field generated by a root θ of a monic irreducible trinomial <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:mi>F</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>+</m:mo> <m:mi>a</m:mi> <m:msup> <m:mi>x</m:mi> <m:mi>m</m:mi> </m:msup> <m:mo>+</m:mo> <m:mi>b</m:mi> <m:mo>∈</m:mo> <m:mi>ℤ</m:mi> <m:mrow> <m:mo>[</m:mo> <m:mi>x</m:mi> <m:mo>]</m:mo> </m:mrow> </m:mrow> </m:math> . In this paper, we study the problem of monogenity of K . More precisely, we provide some explicit conditions on a, b, n , and m for which K is not monogenic. As applications, we show that there are infinite families of non-monogenic number fields defined by trinomials of degree n = 2 r · 3 k with r and k two positive integers. We also give infinite families of non-monogenic sextic number fields defined by trinomials. Some illustrating examples are giving at the end of this paper.