
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 and over a filed . We show that the rings and are isomorphic if and only if their residue fields and 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 , we prove that we have an isomorphism between and .
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Taxonomy
TopicsRings, Modules, and Algebras · Advanced Differential Equations and Dynamical Systems · Advanced Topics in Algebra
