Document Type : Original Article

Authors

• V. Parthiban ¹
• C. Aruna ²

¹ VIT Chennai

² VIT University Chennai, India

Abstract

We investigate two parametric families of irreducible sextic polynomials over $\mathbb{Q}$, 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
\[
K_t = \mathbb{Q}(\alpha) \quad \text{and} \quad L_t = \mathbb{Q}(\theta),
\]
where $\alpha$ and $\theta$ 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 $K_t$ contains real quadratic subfields of the form
\[
\mathbb{Q}(\sqrt{t^2 - 4}),
\]
and that every real quadratic field embeds in some $K_t$. Meanwhile, each $L_t$ contains a cubic subfield of the form
\[
\mathbb{Q}(\theta^2 - \theta).
\]

These results suggest that exceptional and monogenic number fields arise naturally and frequently in well-structured parametric families.

Keywords

• Exceptional units
• sextic fields
• monogenicity

Main Subjects

• Algebraic number theory (MSC 11R, 11S)