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On the simplest quartic fields and related Thue equations

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.