Exceptional Quartics are Ubiquitous
Aruna C, P Vanchinathan

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 , and their normal closu res have Galois groups and 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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Polynomial and algebraic computation · Commutative Algebra and Its Applications
