Extended Eisenstein's Criterion
Abstract
This research paper gives an in-depth analysis of Eisenstein’s criterion, as an essential tool in determining the irreducibility of polynomials over the rational numbers. It begins with exploring the theoretical foundations of the criterion and its classical conditions involving divisibility by a prime number. According to these foundations, we introduce two distinct methodological extensions for applying this criterion. The first extension, known as the direct method, investigates the suggestions of assuming that the chosen prime number divides particular non-prime components within the polynomial. The second extension, described as an indirect method, involves analyzing the polynomial’s behavior under the exchange of specific prime values. Using these two methods, this paper explores the Eisenstein criterion as a domain within the bounds of possible generalizations. The results thus provide new insights into the criterion's applicability and applications, contributing to a deeper understanding of the irreducibility of many algebraic contexts.
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