2023/10/19 by Alex Feiner, Feiner, Alex, Zion Hefty +1
Computer Science · Mathematics · #06F05 #11D07 #11H06 #Algebraic Geometry and Number Theory #Coding theory and cryptography #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2310.12530
openalex publication_date 2023/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a number field K that is a subfield of the real numbers, we generalize the notion of the classical Frobenius problem to the ring of integers \mathfrakOK of K by describing certain Frobenius semigroups, Frob(α1,…,αn), for appropriate elements α1,…,αn∈\mathfrakOK. We construct a partial ordering on Frob(α1,…,αn), and show that this set is completely described by the maximal elements with respect to this ordering. We also show that Frob(α1,…,αn) will always have finitely many such maximal elements, but in general, the number of maximal elements can grow without bound as n is fixed and α1,…,αn∈\mathfrakOK vary. Explicit examples of the Frobenius semigroups are also calculated for certain cases in real quadratic number fields.