In this chapter, we lay some algebraic foundations for a systematic study of curves, specifically the question of factorizing polynomials in one or more variables. Later, we will return to more special topics in algebra, but till then this material will serve all our needs. We start by discussing the abstract concept of factorization in an arbitrary domain. In the succeeding section, we recall the elementary algebra associated to polynomials in a single variable; that provides a model for the algebra of polynomials in several variables, and is anyway of particular relevance to the geometry of curves. We devote a separate section to the special polynomials known as ‘forms’, crucial to understanding ‘tangents’ and ‘projective’ curves. Finally, we explain how some basic calculus ideas can be introduced for polynomials over arbitrary domains.

Factorization in Domains

Much of this book will centre around the concept of ‘factorizing’ polynomials. The domain ℤ of integers provides the model for the ideas, but we phrase the definitions for any domain D. Given elements a, b ∈ D we say that a is a factor of b, or that b is divisible by a, written a|b, when there exists an element c ∈ D with b = ac. (Note that according to this definition the zero element 0 is divisible by any element in D.) A factor is said to be trivial when it is a unit, otherwise it is proper.