Local Euler characteristic formula

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".
Statement.
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, "Hi"("K", "M") = 0 for "i" ≥ 3. Therefore, the Euler characteristic only involves the groups with "i" = 0, 1, 2.
Case of finite modules.
Let "M" be a "GK"-module of finite order "m". The Euler characteristic of "M" is defined to be
(the "i"th 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
i.e. the inverse of the order of the quotient ring "R"/"mR".
Two special cases worth singling out are the following. If the order of "M" is relatively prime to the characteristic of the residue field of "K", then the Euler characteristic is one. If "K" is a finite extension of the "p"-adic numbers Q"p", and if "vp" denotes the "p"-adic valuation, then
where ["K":Q"p"] is the degree of "K" over Q"p".
The Euler characteristic can be rewritten, using local Tate duality, as
where "M"′ is the local Tate dual of "M".