Galois Theory, 5th Edition

Chapter 4Field Extensions

DOI: 10.1201/9781003213949-4

Galois's original theory was couched in terms of polynomials over the complex field. 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 ‘field extension’ related to a polynomial. Every polynomial f over a field K defines another field L containing K (or at any rate a subfield isomorphic to K). There are conceptual advantages in setting up the theory from this point of view. In this chapter we define field extensions and explain the link with polynomials. To keep the discussion concrete ...