first half of the book is modern in its consistent use of adelic and idèlic methods and the simultaneous treatment of algebraic number fields and rational function fields over finite fields. The second half is arguably pre-modern in its development of simple algebras and class field theory without the language of cohomology, and without the language of Galois cohomology in particular. The author acknowledges this as a trade-off, explaining that “to develop such an approach systematically would have meant loading a great deal of unnecessary machinery on a ship which seemed well equipped for this particular voyage; instead of making it more seaworthy, it might have sunk it.” The treatment of class field theory uses analytic methods on both commutative fields and simple algebras. These methods show their power in giving the first unified proof that if K/k is a finite normal extension of A-fields, then any automorphism of K over k is induced by the Frobenius automorphism for infinitely many places of K. This approach also allows for a significantly simpler and more logical proof of algebraic statements, for example the result that a simple algebra over an A-field splits (globally) if and only if it splits everywhere locally. The systematic use of simple algebras also simplifies the treatment of local class field theory. For instance, it is more straightforward to prove that a simple algebra over a local field has an unramified splitting field than to prove the corresponding statement for 2-cohomology classes. Chapter I The book begins with Witt’s formulation of Wedderburn’s proof that a finite division ring is commutative ('Wedderburn's little theorem'). Properties of Haar measure are used to prove that `local fields’ (commutative fields locally compact under a non-discrete topology) are completions of A-fields. In particular – a concept developed later – they are precisely the fields whose local class field theory is needed for the global theory. The non-discrete non-commutative locally compact fields are then division algebras of finite dimension over a local field. Chapter II Finite-dimensional vector spaces over local fields and division algebras under the topology uniquely determined by the field's topology are studied, and lattices are defined topologically, an analogue of Minkowski's theorem is proved in this context, and the main theorems about character groups of these vector spaces, which in the commutative one-dimensional case reduces to `self duality’ for local fields, are shown. Chapter III Tensor products are used to