  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 [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).
