the proofs, we get a canonical generator of H2(E/F), called the fundamental class. It follows from this that the Brauer group H2(E/*) is (canonically) isomorphic to the group Q/Z, except in the case of the archimedean local fields R and C when it has order 2 or 1. Tate's theorem and the Artin map Tate's theorem in group cohomology is as follows. Suppose that A is a module over a finite group G and a is an element of H2(G,A), such that for every subgroup E of G H1(E,A) is trivial, and H2(E,A) is generated by Res(a) which has order E. Then cup product with a is an isomorphism Hn(G,Z) → Hn+2(G,A). If we apply the case n=−2 of Tate's theorem to a class formation, we find that there is an isomorphism H−2(E/F,Z) → H0(E/F,AF) for any normal layer E/F. The group H−2(E/F,Z) is just the abelianization of E/F, and the group H0(E/F,AF) is AE modulo the group of norms of AF. In other words, we have an explicit description of the abelianization of the Galois group E/F in terms of AE. Taking the inverse of this isomorphism gives a homomorphism AE → abelianization of E/F, and taking the limit over all open subgroups F gives a homomorphism AE → abelianization of E, called the Artin map. The Artin map is not necessarily surjective, but has dense image. By the existence theorem below its kernel is the connected component of AE (for class field theory), which is trivial for class field theory of non-archimedean local fields and for function fields, but is non-trivial for archimedean local fields and number fields. The Takagi existence theorem The main remaining theorem of class field theory is the Takagi existence theorem, which states that every finite index closed subgroup of the idele class group is the group of norms corresponding to some abelian extension. The classical way to prove this is to construct some extensions with small groups of norms, by first adding in many roots of unity, and then taking Kummer extensions and Artin–Schreier extensions. These extensions may be non-abelian (though they are extensions of abelian groups by abelian groups); however, this does not really matter, as the norm group of a non-abelian Galois extension is the same as that of its maximal abelian extension (this can be shown using what we already know about class fields). This gives enough (abelian) extensions to