Exceptional units and Euclidean number fields

Abstract

By a result of H.W. Lenstra, one can prove that a number field is Euclidean with the aid of exceptional units. We describe two methods computing exceptional sequences, i.e., sets of units such that the difference of any two of them is still a unit. The second method is based on a graph theory algorithm for the maximum clique problem. This yielded 42 new Euclidean number fields in degrees 8, 9, 10, 11 and 12.

On the sumsets of exceptional units in $$\mathbb {Z}_n$$ Z n

“…After that, many new Euclidean number fields were found (see [6,9]). Moreover, exceptional units also have connections with cyclic resultants [18,19] and Lehmer's conjecture related to Mahler's measure [15,16].…”

“…In 1977, by using exceptional units, Lenstra [5] introduced a method to find Euclidean number fields. After that, many new Euclidean number fields were found (See [4] and [6]). Beyond these, exceptional units also have connections with cyclic resultants [15,16] and Lehmer's conjecture related to Mahler's measure [12,13].…”

Sums of exceptional units in residue class rings

“…By further development of this method, quite a few formerly unknown Euclidean number fields could be found by Leutbecher and Niklasch [5] and Houriet [3]. Exunits were also studied for their own sake, e.g., the calculation of the number of exunits in a number field of given degree and unit rank [7].…”

On the sums of squares of exceptional units in residue class rings

“…Lenstra [6] used exceptional units to find Euclidean number fields. Since then, many new Euclidean number fields were found, see, for example, [4,9]. Besides, exceptional units are connected with the investigation of cyclic resultants [16,17], Salem numbers and Lehmer's conjecture related to Mahler's measure [13,14].…”