On the Field Isomorphism Problem for the family of simplest quartic fields
Volume 218 / 2025
Acta Arithmetica 218 (2025), 347-356
MSC: Primary 11R16; Secondary 11B37
DOI: 10.4064/aa240619-15-10
Published online: 31 January 2025
Abstract
Deciding whether or not two polynomials have isomoprhic splitting fields over the rationals is the Field Isomorphism Problem. We consider polynomials of the form $f_n(x) = x^4-nx^3-6x^2+nx+1$ with $n \neq 3$ a positive integer and we let $K_n$ denote the splitting field of $f_n(x)$, a ‘simplest quartic field’. Our main theorem states that under certain hypotheses there can be at most one positive integer $m \neq n$ such that $K_m=K_n$. The proof relies on the existence of squares in recurrent sequences and a result of J. H. E. Cohn (1972). These sequences allow us to establish uniqueness of the splitting field under additional hypotheses and to establish a connection with elliptic curves.
