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.
Highlights
Biquadratic fields are numerical fields that have been studied extensively [1,2,3,4,5,6] and are currently in the spotlight as they provide examples of non-principal Euclidean ideal classes [7,8]. They are defined as numerical fields whose Galois group is the Klein group, and they can be obtained by the compositum of two quadratic fields, which is the construction we adopt in the present work
Among all the prime ideals, those of degree one have seen a fruitful employment in computational number theory. These first-degree prime ideals may be represented using elementary finite arithmetic and are well suited for practical computing. This feature is crucial for the effectiveness of the general number field sieve (GNFS), which is the most efficient known algorithm to factorize large integers [10,11,12]
We start from a biquadratic field Q(γ), and we investigate the relation between the first-degree prime ideals in Z[γ] and the first-degree prime ideals of Z[α] and Z[ β], with Q(α) and
Summary
Biquadratic fields are numerical fields that have been studied extensively [1,2,3,4,5,6] and are currently in the spotlight as they provide examples of non-principal Euclidean ideal classes [7,8]. The study of the prime ideals of number field orders and conditions on the ideals they divide are common and often challenging goals in commutative algebra These topics find a natural application in algebraic number theory, and they have been recently widely adopted for addressing concrete computational problems, such as integer factorization [9]. These first-degree prime ideals may be represented using elementary finite arithmetic and are well suited for practical computing This feature is crucial for the effectiveness of the general number field sieve (GNFS), which is the most efficient known algorithm to factorize large integers [10,11,12]. With their application to the GNFS in mind, we determine conditions under which these special primes in the biquadratic extension divide prescribed principal ideals in terms of their behavior in the quadratic extensions.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
