the integers. Local class field theory of characteristic p>0: The module A is the separable algebraic closure of the field of formal Laurent series over a finite field, and G is the Galois group. Non-archimedean local class field theory of characteristic 0: The module A is the algebraic closure of a field of p-adic numbers, and G is the Galois group. Global class field theory of characteristic p>0: The module A is the union of the groups of idele classes of separable finite extensions of some function field over a finite field, and G is the Galois group. Global class field theory of characteristic 0: The module A is the union of the groups of idele classes of algebraic number fields, and G is the Galois group of the rational numbers (or some algebraic number field) acting on A. It is easy to verify the class formation property for the finite field case and the archimedean local field case, but the remaining cases are more difficult. Most of the hard work of class field theory consists of proving that these are indeed class formations. This is done in several steps, as described in the sections below. The first inequality The first inequality of class field theory states that |H0(E/F)| ≥ |E/F| for cyclic layers E/F. It is usually proved using properties of the Herbrand quotient, in the more precise form |H0(E/F)| = |E/F|×|H1(E/F)|. It is fairly straightforward to prove, because the Herbrand quotient is easy to work out, as it is multiplicative on short exact sequences, and is 1 for finite modules. Before about 1950, the first inequality was known as the second inequality, and vice versa. The second inequality The second inequality of class field theory states that |H0(E/F)| ≤ |E/F| for all normal layers E/F. For local fields, this inequality follows easily from Hilbert's theorem 90 together with the first inequality and some basic properties of group cohomology. The second inequality was first proved for global fields by Weber using properties of the L series of number fields, as follows. Suppose that the layer E/F corresponds to an extension k⊂K of global fields. By studying the Dedekind zeta function of K one shows that the degree 1 primes of K have Dirichlet density given by the order of the pole at s=1, which is 1 (When K is the rationals, this is essentially Euler's proof that there are