ABSTRACT
Galois's original theory was couched in terms of polynomials over the complex numbers. The modern approach is a consequence of the methods used, starting around 1890 and flourishing in the 1920s and 1930s, to generalise the theory to arbitrary fields. From this viewpoint the central object of study ceases to be a polynomial, and becomes instead a related field extension. Every polynomial over a field K defines another field L containing K. There are conceptual advantages in setting up the theory from this point of view. In this chapter we define field extensions (always working inside C) and explain the link with polynomials.
