Cyclotomic Function Fields

Abstract

In the last chapter we explored the arithmetic of constant field extensions and noted (as was pointed out by Iwasawa) that these extensions can be thought of as function field analogues of cyclotomic extensions of number fields. This analogy led to various conjectures about the behavior of class groups in number fields which have proved very fruitful for the development of algebraic number theory and arithmetic geometry. There is another function field analogy to cyclotomic number fields which was first discovered by L. Carlitz [3] in the late 1930s. This ingenious analogy was not well known until D. Hayes, in 1973, published an exposition of Carlitz’s idea and showed that it provided an explicit class field theory for the rational function field (see Hayes [1]). Later developments, due independently to Hayes and V. Drinfeld, showed that Carlitz’s ideas can be generalized to provide an explicit class field theory for any global function field, i.e., an explicit construction of all abelian extensions of such a field (see Drinfeld [1] and Hayes [2]). This is a complete solution to Hilbert’s 9-th problem in the function field case. Nothing remotely so satisfying is known for number fields except for the field of rational numbers (cyclotomic theory) and imaginary quadratic number fields (the theory of complex multiplication).