  In this article we give an adelic proof of the Chevalley-Gras formula for
global fields, which itself is a generalization of the ambiguous class number
formula. The idea is to reduce the formula to the Hasse norm theorem, the local
and global reciprocity laws. We also give an adelic proof of the Chevalley-Gras
formula for $0$-th divisor class group in the function field case, which
extends a result of Rosen.
