all normal layers, but this takes more work; see the next section.) The Brauer group The Brauer groups H2(E/*) of a class formation are defined to be the direct limit of the groups H2(E/F) as F runs over all open subgroups of E. An easy consequence of the vanishing of H1 for all layers is that the groups H2(E/F) are all subgroups of the Brauer group. In local class field theory the Brauer groups are the same as Brauer groups of fields, but in global class field theory the Brauer group of the formation is not the Brauer group of the corresponding global field (though they are related). The next step is to prove that H2(E/F) is cyclic of order exactly |E/F|; the previous section shows that it has at most this order, so it is sufficient to find some element of order |E/F| in H2(E/F). The proof for arbitrary extensions uses a homomorphism from the group G onto the profinite completion of the integers with kernel G∞, or in other words a compatible sequence of homomorphisms of G onto the cyclic groups of order n for all n, with kernels Gn. These homomorphisms are constructed using cyclic cyclotomic extensions of fields; for finite fields they are given by the algebraic closure, for non-archimedean local fields they are given by the maximal unramified extensions, and for global fields they are slightly more complicated. As these extensions are given explicitly one can check that they have the property that H2(G/Gn) is cyclic of order n, with a canonical generator. It follows from this that for any layer E, the group H2(E/E∩G∞) is canonically isomorphic to Q/Z. This idea of using roots of unity was introduced by Chebotarev in his proof of Chebotarev's density theorem, and used shortly afterwards by Artin to prove his reciprocity theorem. For general layers E,F there is an exact sequence The last two groups in this sequence can both be identified with Q/Z and the map between them is then multiplication by |E/F|. So the first group is canonically isomorphic to Z/nZ. As H2(E/F) has order at most Z/nZ is must be equal to Z/nZ (and in particular is contained in the middle group)). This shows that the second cohomology group H2(E/F) of any layer is cyclic of order |E/F|, which completes the verification of the axioms of a class formation. With a little more care in