study extensions of the places of an A-field to places of a finite separable extension of the field, with the more complicated inseparable case postponed to later. Chapter IV This chapter introduces the topological adele ring and idèle group of an A-field, and proves the `main theorems’ as follows: both the adele ring and the idèle group are locally compact; the A-field, when embedded diagonally, is a discrete and co-compact subring of its adele ring; the adele ring is self dual, meaning that it is topologically isomorphic to its Pontryagin dual, with similar properties for finite-dimensional vector spaces and algebras over local fields. The chapter ends with a generalized unit theorem for A-fields, describing the units in valuation terms. Chapter V This chapter departs slightly from the simultaneous treatment of number fields and function fields. In the number field setting, lattices (that is, fractional ideals) are defined, and the Haar measure volume of a fundamental domain for a lattice is found. This is used to study the discriminant of an extension. Chapter VI This chapter is focused on the function field case; the Riemann-Roch theorem is stated and proved in measure-theoretic language, with the canonical class defined as the class of divisors of non-trivial characters of the adele ring which are trivial on the embedded field. Chapter VII The zeta and L-functions (and similar analytic objects) for an A-field are expressed in terms of integrals over the idèle group. Decomposing these integrals into products over all valuations and using Fourier transforms gives rise to meromorphic continuations and functional equations. This gives, for example, analytic continuation of the Dedekind zeta-function to the whole plane, along with its functional equation. The treatment here goes back ultimately to a suggestion of Artin, and was developed in Tate’s thesis. Chapter VIII Formulas for local and global differents and discriminants, ramification theory, and the formula for the genus of an algebraic extension of a function field are developed. Chapter IX A brief treatment of simple algebras is given, including explicit rules for cyclic factor sets. Chapters X and XI The zeta-function of a simple algebra over an A-field is defined, and used to prove further results on the norm group and groupoid of maximal ideals in a simple algebra over an A-field. Chapter XII The reciprocity law of local class field theory over a local field in the context of a pairing of the multiplicative