Constructing infinite families of number fields with fixed indices via sextinomials

Abstract In this paper, we investigate the p -adic valuation ν p ( i ( K )) of the index i ( K ) of an algebraic number field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>K</m:mi> <m:mo>=</m:mo> <m:mi mathvariant="double-struck">Q</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> $K=\mathbb{Q}(\theta )$ , where θ is a root of an irreducible sextinomial of the type <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mi>a</m:mi> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mi>e</m:mi> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mi>b</m:mi> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mi>c</m:mi> <m:mi>x</m:mi> <m:mo>+</m:mo> <m:mi>d</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">Z</m:mi> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:math> ${x}^{n}+a{x}^{m}+e{x}^{3}+b{x}^{2}+cx+d\in \mathbb{Z}[x]$ . For a given rational prime p and certain natural numbers i p , we provide infinite families of number fields K for which ν p ( i ( K )) = i p . In particular, i 2 ∈ {1, 2, 3, 4, 5, 6, 7, 8}, while for every odd prime p , we construct families for which i p ∈ {1, 2, 3, p − 2, p − 1, p , p + 2}. Several explicit examples are provided to illustrate the theoretical results.