First-Degree Prime Ideals of Biquadratic Fields Dividing Prescribed Principal Ideals

Abstract

We describe first-degree prime ideals of biquadratic extensions in terms of the first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms. The correspondence between these ideals in the larger ring and those in the smaller ones extends to the divisibility of specially-shaped principal ideals in their respective rings, with some exceptions that we explicitly characterize.

First-degree prime ideals of composite extensions

“…The employed hypotheses are not truly restrictive: every pair of reasonably uncorrelated fields happens to be linearly disjoint [10,17], thus every composite extension may be realized this way, with a suitable choice of sub-extensions. The exceptional cases are precisely identified, and ad hoc examples are provided to show that every given hypothesis is essential.This work extends a previous work of the authors [22], which addresses this problem when the considered number fields are biquadratic. However, the techniques employed and developed in the current paper are more sophisticated and lead to a deeper comprehension of the involved objects.…”

“…This work extends a previous work of the authors [22], which addresses this problem when the considered number fields are biquadratic. However, the techniques employed and developed in the current paper are more sophisticated and lead to a deeper comprehension of the involved objects.…”

“…However, the techniques employed and developed in the current paper are more sophisticated and lead to a deeper comprehension of the involved objects. The novel results not only generalize those of [22], but also provide practical tools that may be applied to a much wider range of situations, such as those that are conventionally adopted for the GNFS implementation, namely, nowadays, degree-6 extensions.This paper is organized as follows: in Section 2 the basic results about resultant and linearly disjoint extensions are recalled and combined to identify the number fields considered in the present work. Section 3 is devoted to defining the first-degree prime ideals combination and to detailing the cases when this construction establishes a complete correspondence with the first-degree prime ideals of the sub-fields.…”

“…However, the techniques employed and developed in the current paper are more sophisticated and lead to a deeper comprehension of the involved objects. The novel results not only generalize those of [22], but also provide practical tools that may be applied to a much wider range of situations, such as those that are conventionally adopted for the GNFS implementation, namely, nowadays, degree-6 extensions.…”

“…The division of ideals in Z[θ] using only first-degree prime ideals is completely addressed in [7], as it is one of the main tools on which the GNFS relies. See also [22,Section 2] for a quick recap on these properties.…”