An infinite series of Kronecker conjugate polynomials

Abstract:

Let $K$ be a field of characteristic $0$, $t$ a transcendental over $K$, and $\Gamma$ be the absolute Galois group of $K(t)$. Then two non-constant polynomials $f,g\in K[X]$ are said to be Kronecker conjugate if an element of $\Gamma$ fixes a root of $f(X)-t$ if and only if it fixes a root of $g(X)-t$. If $K$ is a number field, and $f,g\in {\mathcal O}_K[X]$ where ${\mathcal O}_K$ is the ring of integers of $K$, then $f$ and $g$ are Kronecker conjugate if and only if the value set $f({\mathcal O}_K)$ equals $g({\mathcal O}_K)$ modulo all but finitely many non-zero prime ideals of ${\mathcal O}_K$. In 1968 H. Davenport suggested the study of this latter arithmetic property. The main progress is due to M. Fried, who showed that under certain assumptions the polynomials $f$ and $g$ differ by a linear substitution. Further, he found non-trivial examples where Kronecker conjugacy holds. Until now there were only finitely many known such examples. This paper provides the first infinite series. The main part of the construction is group theoretic.