2024/06/14 by David L. Pincus, Pincus, David L., Lawrence C. Washington +1
Computer Science · #11B37 #11R16 #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2406.10414
openalex publication_date 2024/06/14 · openalex created_date 2024/06/19 · openalex updated_date 2026/07/28
Deciding whether or not two polynomials have isomoprhic splitting fields over the rationals is the Field Isomorphism Problem. We consider polynomials of the form fn(x) = x4-nx3-6x2+nx+1 with n ≠ 3 a positive integer and we let Kn denote the splitting field of fn(x); a `simplest quartic field'. Our main theorem states that under certain hypotheses there can be at most one positive integer m ≠ n such that Km=Kn. The proof relies on the existence of squares in recurrent sequences and a result of J.H.E. Cohn [3]. These sequences allow us to establish uniqueness of the splitting field under additional hypotheses in Section (5) and to establish a connection with elliptic curves in Section (6).