On some local rings

TL;DR

This paper investigates isomorphisms between certain local rings derived from separable irreducible polynomials over a field, establishing conditions under which these rings are isomorphic, and exploring related algebraic structures.

Contribution

It provides a characterization of when rings formed by quotienting polynomial rings by powers of separable irreducible polynomials are isomorphic, extending understanding in algebraic ring theory.

Findings

Isomorphism between [X]/(P_1^n) and [X]/(P_2^n) iff their residue fields are isomorphic

Established an isomorphism between [X]/(P^n) and ([X](P))[Y]/(Y^n) for separable irreducible P

Partial results for non-separable polynomials

Abstract

Given two seprable irreducible polynomials $P_1$ and $P_2$ over a filed $\mathbb{K}$. We show that the rings $\mathbb{K}[X]/(P_1^n)$ and $\mathbb{K}[X]/(P_2^n)$ are isomorphic if and only if their residue fields $\mathbb{K}[X]/(P_1)$ and $\mathbb{K}[X]/(P_2)$ are isomorphic. Partial results in this direction are obtained for the case where the polynomials are not seprable. We note that, given a seprable irreducible polynomial $P$, we prove that we have an isomorphism between $\mathbb{K}[X]/(P^n)$ and $(\mathbb{K}[X](P))[Y]/(Y^n)$.