Delving Deeper: Decomposing Quintics

Abstract

Polynomial decomposition, also referred to as polynomial factorization, is the process of splitting a given polynomial of degree n into its constituent factors—that is, polynomials of lower degree. A reducible polynomial over a given field—such as the real (ℝ), complex (ℂ), or rational numbers (ℚ)—is one that can be factored into polynomials of lower degree with coefficients in that field; otherwise, it is irreducible over the field (Thangadurai 2007). From the fundamental theorem of algebra, we know that a polynomial with integer coefficients is completely reducible into linear factors over the complex field, whereas it is reducible to linear and quadratic factors over ℝ; however, it may be irreducible over ℚ. If a polynomial (with rational coefficients) can be factored over ℚ, Gauss's lemma states, it can be factored over the integers as well. The fundamental theorem of algebra is only an existence proof and does not provide any procedure for factoring a polynomial.