Exceptional Quartics are Ubiquitous

TL;DR

This paper constructs infinite families of exceptional quartic number fields with various properties, including containing quadratic subfields, being non-Galois, and having power integral bases, extending to higher degrees.

Contribution

It provides explicit constructions of infinitely many exceptional quartic fields with diverse Galois groups and monogenic properties, and extends the concept to higher degrees.

Findings

Existence of infinitely many exceptional quartic fields containing a given quadratic field.

Existence of infinitely many exceptional quartic fields without quadratic subfields.

Many of these fields are monogenic and have specified Galois groups.

Abstract

For each real quadratic field we constructively show the existence of infinitely many exceptional quartic number fields containing that quadratic field. On the other hand, another infinite collection of quartic exceptional fields without any quadratic subfields is also provided. Both these families are non-Galois extensions of $\mathbf{Q}$, and their normal closu res have Galois groups $D_4$ and $S_4$ respectively. We also show that an infinite number of these exceptional quartic fields have power integral basis, i.e., monogenic. We also construct large collections of exceptional number fields in all degrees greater than 4.