A Method of Finding Irreducible Polynomials

This paper introduces a simplified approach to finding irreducible polynomials, a fundamental concept in abstract algebra crucial for understanding field and ring structures. Traditionally, identifying these polynomials involves complex computations or heuristic methods. Our study presents a straightforward method by constructing and proving the integrality of the polynomial roots, bridging advanced and elementary mathematical principles. The proposed method utilizes Vietas formulas and roots of unity to systematically construct potential roots of an element's irreducible polynomial across various fields. We provide elementary proofs using induction and polynomial properties, demonstrating the methods effectiveness through examples in both rational and finite fields.