Ideal Theory and Algebraic Curves

Abstract

Polynomials in two variables define algebraic curves in the plane, and algebraic curves in the plane generally meet (perhaps in complicated ways) in points. What is the connection between the geometry and the algebra? In this chapter we shall see how this question was answered, not entirely successfully, in the late nineteenth century by two mathematicians: Alexander Brill and Max Noether (the father of the more illustrious Emmy). The generalisation to more variables was very difficult, and was chiefly the achievement of Emanuel Lasker, who was the World Chess champion at the time, with his theory of primary ideals. We shall give an example of his fundamental result taken from the English mathematician F.S. Macaulay’s fundamental work on polynomial rings. With these results, the basic structural features of polynomial rings were all in place, and with the equally rich theory of number fields so too were all the basic features of commutative algebra.