Kronecker conjugacy of polynomials

Müller, Peter

doi:10.1090/S0002-9947-98-02123-0

Kronecker conjugacy of polynomials
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Trans. Amer. Math. Soc. 350 (1998), 1823-1850 Request permission

Abstract:

Let $f,g\in \mathbb {Z}[X]$ be non-constant polynomials with integral coefficients. In 1968 H. Davenport raised the question as to when the value sets $f(\mathbb {Z})$ and $g(\mathbb {Z})$ are the same modulo all but finitely many primes. The main progress until now is M. Fried’s result that $f$ and $g$ then differ by a linear substitution, provided that $f$ is functionally indecomposable. We extend this result to polynomials $f$ of composition length $2$. Also, we study the analog when $\mathbb {Z}$ is replaced by the integers of a number field. The above number theoretic property translates to an equivalent property of subgroups of a finite group, known as Kronecker conjugacy, a weakening of conjugacy which has been studied by various authors under different assumptions and in other contexts. We also give a simplified and strengthened version of the Galois theoretic translation to finite groups.

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