On index divisors of certain sextic number fields defined by quadrinomials

Abstract Let K be a number field of degree 6 generated by a root of an irreducible quadrinomial <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>F</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi>x</m:mi> <m:mn>6</m:mn> </m:msup> <m:mo>+</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>m</m:mi> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>b</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>x</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:mi>c</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> F(x)=x^{6}+ax^{m}+bx^{2}+c\in{\mathbb{Z}}[x] with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>3</m:mn> <m:mo>,</m:mo> <m:mn>4</m:mn> <m:mo>,</m:mo> <m:mn>5</m:mn> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> {m\in\{3,4,5\}} , and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {i(K)} denote the index of K . In this paper, for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {p=2} or 3, we give sufficient conditions on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>b</m:mi> </m:mrow> </m:math> {a,b} and c such that p is a common index divisor of K , and we evaluate <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>ν</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\nu_{p}(i(K))} , the highest power of p dividing <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {i(K)} . In this way, we provide a partial answer to Problem 22 of [W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, 3rd ed., Springer Monogr. Math., Springer, Berlin, 2004] for these families of number fields. As an application of our results, we identify new classes of sextic number fields having no power integral basis. Further, several corollaries and examples are provided which illustrate our results. Our approach is based on a theorem of Ore on the decomposition of primes in number fields. [18, 30, 34].