The minimum discriminant of number fields of degree 8 and signature (2,3)

Abstract

In this paper we describe how to use the algorithmic methods provided by Hunter and Pohst in order to give a complete classification of number fields of degree 8 and signature (2, 3) with absolute discriminant less than a certain bound. The choice of this bound comes from the local corrections given by prime ideals to the lower estimates for discriminants obtained with the Odlyzko-Poitou-Serre method.

A conjectural improvement for inequalities related to regulators of number fields

“…It seems however that (6, 1) is the only signature in degree 8 for which one can obtain results by the conjectural estimates. Surely signature (2,3) is not affected because we know that M (8, 3) = 8 8/2 by Lemma 2, so that in this case we are still stuck with the previous estimate given by Remak-Friedman's inequality. For what concerns signature (4, 2), we would have an improvement given by using the (conjectured) correct value 2 log M (8, 2) = 2 log 7 7/2 = 7 log 7 instead of the upper bound 8 log 8.…”

“…The missing signature (5, 1) in degree 7 was solved later by Friedman and Ramirez-Raposo [10] with an ad hoc improvement to Remak-Friedman's inequality which allowed to implement the procedure. Thus, next cases in which a classification of this kind is not yet known are the remaining signatures (2,3), (4,2) and (6,1) in degree 8. One of the reasons why this study was skipped by the previous authors was the lack of complete tables of number fields up to some discriminant bounds, which prove to be crucial to guarantee the correctness of the procedure.…”

“…One of the reasons why this study was skipped by the previous authors was the lack of complete tables of number fields up to some discriminant bounds, which prove to be crucial to guarantee the correctness of the procedure. We were able to provide such lists for these signatures [2,3], and we tried to apply the classification method on the considered signatures; the attempt however was not successful, and the reasons are similar to the ones which prevented the previous authors to immediately solve the case of signature (5,1).…”

“…Let (r 1 , r 2 ) be one of these signatures. From [2,3] one gets complete tables of fields K ∈ F r 1 ,r 2 with |d K | ≤ d 1 , where d 1 depends on the signature and is specified in Table 1. For every field in the list, we compute their regulators R K using PARI and we verify that the output satisfies (5).…”

On small discriminants of number fields of degree 8 and 9

“…For what concerns further signatures in degree 8, no complete tables up to some bound were known and for several years no attempts of this kind were made. During his Ph.D. work, the author [3] was then able to give a complete classification of number fields with degree 8, signature (2, 3) and |d K | ≤ 5726301, showing that there exist exactly 56 such fields: this result was obtained by exploiting the aforementioned theoretical ideas in order to write an algorithmic procedure which was implemented in a program relying on the softwares MATLAB and PARI/GP [38]. This setting was not good enough for exploring other signatures in degree 8 and 9, but we are now able to provide a better implementation, needing just PARI/GP, which allowed us to obtain the following classification result.…”