On the Field Isomorphism Problem for the Family of Simplest Quartic Fields

Deciding whether or not two polynomials have isomoprhic splitting fields over the rationals is the Field Isomorphism Problem. We consider polynomials of the form $f_n(x)=x^4-n{x}^3-6x^2+nx+1$ with $n\mathrm{\ne }3$ 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 $m\mathrm{\ne }n$ 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 [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).