We continue our quest to climb Mount Takagi-Artin.

In class field theory, it will be important to compute and keep track of the orders of groups such as , where is a Galois extension of local fields. A convenient piece of machinery for doing this is the Herbrand quotient, which we discuss today. I only sketch the proofs though, and a little familiarity with the Tate cohomology groups will be useful (but is not strictly necessary if one accepts the essentially combinatorial results without proof or proves them directly).

**1. Definition **

Let be a cyclic group generated by and a -module. It is well-known that the Tate cohomology groups are periodic with period two and thus determined by and . By definition,

where consists of the elements of fixed by , and is the norm map, . Moreover,

(Normally, for only assumed finite, we would quotient by the sum of for arbitrary, but here it is enough to do it for a generator—easy exercise.)

If both cohomology groups are finite, define the **Herbrand quotient** as