  Let $K$ be a field of char $K\neq 2$. For $a\in K$, we give an explicit
answer to the field isomorphism problem of the simplest quartic polynomial
$X^4-aX^3-6X^2+aX+1$ over $K$ as the special case of the field intersection
problem via multi-resolvent polynomials. From this result, over an infinite
field $K$, we see that the polynomial gives the same splitting field over $K$
for infinitely many values $a$ of $K$. We also see by Siegel's theorem for
curves of genus zero that only finitely many algebraic integers
$a\in\mathcal{O}_K$ in a number field $K$ may give the same splitting field. By
applying the result over the field $\mathbb{Q}$ of rational numbers, we
establish a correspondence between primitive solutions to the parametric family
of quartic Thue equations \[ X^4-mX^3Y-6X^2Y^2+mXY^3+Y^4=c, \] where
$m\in\mathbb{Z}$ is a rational integer and $c$ is a divisor of $4(m^2+16)$, and
isomorphism classes of the simplest quartic fields.
