ABSTRACT
This chapter explores irreducible polynomials over “classical” fields: the rational numbers, the real numbers, and the complex numbers. It describes the Eisenstein’s irreducibility criterion and the fundamental theorem of algebra. The chapter shows how a relatively simple looking polynomial will factor into irreducibles in completely different ways over five different fields (three infinite and two finite). It also includes exercises related to the concept of polynomials over the rational numbers, the real numbers, and the complex numbers.
