Class Formation - The First Inequality

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 straighforward 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. What is now the 'second' was once the 'first' (see for example p. 49 in this treatment (PDF); this bounds the index of the norms in a class group, in old-fashioned language, and is the part of the main proof that was initially treated by means of L-functions. The historical reason behind this is that the first inequality of genus theory (concerned with 2-torsion in the class groups of quadratic fields) was an upper bound for the number of genera. (discussed at introduction to the Hilbert Zahlbericht (PDF).