Polynomials

Abstract

From a more algebraic point of view, real polynomial functions can be seen as polynomials with real coefficients. As we shall see from this chapter on, such a change of perspective turns out to be quite fruitful, so much that we shall not restrict ourselves to polynomials with real, or even complex, coefficients; later chapters will deal with the case of polynomials with coefficients in $\mathbb Z_p$, for some prime number p. As a result of such generality, we will be able to prove several results on Number Theory which would otherwise remain unaccessible. Our purpose in this chapter is, thus, to start this journey by developing the most elementary algebraic concepts and results on polynomials. To this end, along all that follows we shall write $\mathbb K$ to denote one of $\mathbb Q$,$\mathbb R$ or $\mathbb C$, whenever a specific choice of one of these number sets is immaterial.