On the class number and unit index of simplest quartic fields

Abstract

The term “simplest” field has been used to describe certain totally real, cyclic number fields of degrees 2, 3, 4, 5, 6, and 8. For each of these degrees, the fields are defined by a one-parameter family of polynomials with constant term ±1. The regulator of these “simplest” fields is small in an asymptotic sense: in consequence, the class number of these fields tends to be large.

Elliptic curves associated with simplest quartic fields

“…These fields were studied, among other things, by Gras, who proved that this family is infinite [5]. Later, Lazarus studied their class number [8,9]. More recently, they were studied by Louboutin [10], Kim [7] and Olajos [11].…”

On the use of explicit bounds on residues of Dedekind zeta functions taking into account the behavior of small primes

“…Least values of m ≥ −1 for which ∆m = m 2 + 3m + 9 is prime and h2h3 ≥ ∆m [Laz2] and [Lou04b]) are the real cyclic quartic number fields associated with the quartic polynomials…”

Simple families of Thue inequalities

“…Definition 1 implies that the roots of P generate a cyclic number field of degree deg (F ) , where the Galois action on the roots of P is given by ψ A . These fields are well-known and some of them are the so-called "simplest" number fields (see [22], [9], [8], Appendix of [21]). 4.…”