ABSTRACT
This chapter considers polynomials defined over both rings and fields. It describes a theorem that provides an important result concerning polynomials over a field and their roots. An irreducible polynomial is a polynomial that cannot be factored into polynomials with smaller degrees. Polynomials of degree two or three are irreducible if and only if they do not have a root in the field. The chapter discusses several results concerning the irreducibility of polynomials over the field of real numbers and over the field of complex numbers. It proves that there are irreducible polynomials of degrees one and two over the real numbers and that the only irreducible polynomials over the complex numbers are of degree one. The chapter also presents a few results concerning the problem of solving polynomial equations motivated by the quadratic formula.
