Parametric Constructions of Monogenic and Exceptional Sextic Fields

We investigate two parametric families of irreducible sextic polynomials over ℚ, denoted g(x;t) and h(x;t), and the number fields they generate. For integers t in suitable ranges, we show that the fields Kt = ℚ(α) and Lt = ℚ(θ), where α and θ are roots of g(x;t) and h(x;t), respectively, exhibit rich arithmetic and algebraic structure. In particular, both families define exceptional number fields, and we prove that for infinitely many t, the fields are monogenic. We also show that Kt contains real quadratic subfields of the form ℚ(√t2 - 4), and that every real quadratic field embeds in some Kt. Meanwhile, each Lt contains a cubic subfield of the form ℚ(θ2 - θ).