show that there is an abelian extension corresponding to any finite index subgroup of the idele class group. A consequence is that the kernel of the Artin map is the connected component of the identity of the idele class group, so that the abelianization of the Galois group of F is the profinite completion of the idele class group. For local class field theory, it is also possible to construct abelian extensions more explicitly using Lubin–Tate formal group laws. For global fields, the abelian extensions can be constructed explicitly in some cases: for example, the abelian extensions of the rationals can be constructed using roots of unity, and the abelian extensions of quadratic imaginary fields can be constructed using elliptic functions, but finding an analog of this for arbitrary global fields is an unsolved problem. Weil group This is not a Weyl group and has no connection with the Weil–Châtelet group or the Mordell–Weil group The Weil group of a class formation with fundamental classes uE/F ∈ H2(E/F, AF) is a kind of modified Galois group, introduced by and used in various formulations of class field theory, and in particular in the Langlands program. If E/F is a normal layer, then the Weil group U of E/F is the extension 1 → AF → U → E/F → 1 corresponding to the fundamental class uE/F in H2(E/F, AF). The Weil group of the whole formation is defined to be the inverse limit of the Weil groups of all the layers G/F, for F an open subgroup of G. The reciprocity map of the class formation (G, A) induces an isomorphism from AG to the abelianization of the Weil group. See also Abelian extension Artin L-function Artin reciprocity Class field theory Complex multiplication Galois cohomology Hasse norm theorem Herbrand quotient Hilbert class field Kronecker–Weber theorem Local class field theory Takagi existence theorem Tate cohomology group References , esp. chapter XI: Class formations , reprinted in volume I of his collected papers, Algebraic number theory Class field theory