In the mathematical field of Galois cohomology, the local Euler characteristic formula is a result due to John Tate that computes the Euler characteristic of the group cohomology of the absolute Galois group GK of a non-archimedean local field K. Let K be a non-archimedean local field, let Ks denote a separable closure of K, let GK = Gal(Ks/K) be the absolute Galois group of K, and let Hi(K, M) denote the group cohomology of GK with coefficients in M. Since the cohomological dimension of GK is two,[1] Hi(K, M) = 0 for i ≥ 3.
Therefore, the Euler characteristic only involves the groups with i = 0, 1, 2.
Let M be a GK-module of finite order m. The Euler characteristic of M is defined to be[2] (the ith cohomology groups for i ≥ 3 appear tacitly as their sizes are all one).
Let R denote the ring of integers of K. Tate's result then states that if m is relatively prime to the characteristic of K, then[3] i.e. the inverse of the order of the quotient ring R/mR.
Two special cases worth singling out are the following.