Archimedean local field[edit]
For archimedean local fields the Weil group is easy to describe: for C it is the group C× of non-zero complex numbers, and for R it is a non-split extension of the Galois group of order 2 by the group of non-zero complex numbers, and can be identified with the subgroup C× ∪ j C× of the non-zero quaternions.
Finite field[edit]
For finite fields the Weil group is infinite cyclic. A distinguished generator is provided by the Frobenius automorphism. Certain conventions on terminology, such as arithmetic Frobenius, trace back to the fixing here of a generator (as the Frobenius or its inverse).
Local field[edit]
For a local field of characteristic p > 0, the Weil group is the subgroup of the absolute Galois group of elements that act as a power of the Frobenius automorphism on the constant field (the union of all finite subfields).
For p-adic fields the Weil group is a dense subgroup of the absolute Galois group, and consists of all elements whose image in the Galois group of the residue field is an integral power of the Frobenius automorphism.
More specifically, in these cases, the Weil group does not have the subspace topology, but rather a finer topology. This topology is defined by giving the inertia subgroup its subspace topology and imposing that it be an open subgroup of the Weil group. (The resulting topology is "locally profinite".)
Function field[edit]
For global fields of characteristic p>0 (function fields), the Weil group is the subgroup of the absolute Galois group of elements that act as a power of the Frobenius automorphism on the constant field (the union of all finite subfields).
Number field[edit]
For number fields there is no known "natural" construction of the Weil group without using cocycles to construct the extension. The map from the Weil group to the Galois group is surjective, and its kernel is the connected component of the identity of the Weil group, which is quite complicated.