Automorphisms of Ideals of Polynomial Rings

TL;DR

This paper investigates the automorphism groups of ideals generated by monic polynomials in polynomial rings over integral domains, revealing their structure based on the roots of the polynomial.

Contribution

It characterizes the automorphism groups of ideals generated by monic polynomials, showing they are isomorphic to units or cyclic groups depending on root multiplicity.

Findings

If $f$ has one root, $Aut(I_f) \\cong R^\times$

If $f$ has multiple roots, $Aut(I_f)$ is cyclic

The order of $Aut(I_f)$ is determined by the roots of $f$

Abstract

Let $R$ be a commutative integral domain with unit, $f$ be a nonconstant monic polynomial in $R[t]$, and $I_f \subset R[t]$ be the ideal generated by $f$. In this paper we study the group of $R$-algebra automorphisms of the $R$-algebra without unit $I_f$. We show that, if $f$ has only one root (possibly with multiplicity), then $Aut (I_f) \cong R^\times$. We also show that, under certain mild hypothesis, if $f$ has at least two different roots in the algebraic closure of the quotient field of $R$, then $Aut(I_f)$ is a cyclic group and its order can be completely determined by analyzing the roots of $f$.