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".) Weil group of a function field 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). Weil group of a number field 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. Weil–Deligne group The Weil–Deligne group scheme (or simply Weil–Deligne group) W′K of a non-archimedean local field, K, is an extension of the Weil group WK by a one-dimensional additive group scheme Ga, introduced by . In this extension the Weil group acts on the additive group by where w acts on the residue field of order q as a→a||w|| with ||w|| a power of q. The local Langlands correspondence for GLn over K (now proved) states that there is a natural bijection between isomorphism classes of irreducible admissible representations of GLn(K) and certain n-dimensional representations of the Weil–Deligne group of K. The Weil–Deligne group often shows up through its representations. In such cases, the Weil–Deligne group is sometimes taken to be WK × SL(2,C) or WK × SU(2,R), or is simply done away with and Weil–Deligne representations of WK are used instead. In the archimedean case, the Weil–Deligne group is simply defined to be Weil group. See also Langlands group Shafarevich–Weil theorem Notes References , reprinted in volume I of his collected papers, Class field theory