A correspondence between irregular fields

Correspondences between fields are well known, and Dickson has applied one to obtain a generalization of the theory of numbers. Here we give an instance of correspondence between irregular fields. An irregular field differs from a field only in the exclusion of an infinity of elements as divisors, instead of the uniquely excluded zero of a field. The postulates for an irregular field and numerous instances were given elsewhere. The correspondence is established between the irregular field of all numerical functions and the irregular field of a certain infinity of power series with radius of convergence 1. For the series considered, addition and subtraction are interpreted as in the classical algebra of absolutely convergent series; multiplication and division receive wholly different interpretations. The simplest instance of the new multiplication is the process by which, when legitimate, a Lambert series is derived from a given power series.