In mathematics, a Weil group, introduced by Weil (1951), is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field F, its Weil group is generally denoted WF. There also exists "finite level" modifications of the Galois groups: if E/F is a finite extension, then the relative Weil group of E/F is WE/F = WF/W c
E (where the superscript c denotes the commutator subgroup).
For more details about Weil groups see (Artin & Tate 2009) or (Tate 1979) or (Weil 1951).