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A ring where every nonzero element has an inverse is called a division ring, and we are especially interested in associative structures such as Hamiltons quaternions (since they appear in the icosians context). Commutative division rings are called fields. Due to Wedderburn's theorem every finite division ring is a field.
See [Far93] for a field k we can investigate the (finite-dimensional) central division algebras D over k. The tensor product of two division algebras is not always a division algebra, while the set of central simple algebras is closed under tensor product, so one can put a group structure on the (similarity classes of) central simple k-algebras. This is called the Brauer group Br(k).
Given a field extension K/k, there is a homomorphism
Let S be a central simple k-algebra. A maximal subfield of S is
The groups Hn(G, K*) are the Galois cohomology groups of the extension K / k with coefficients in K*. H0(G, K*) = k* and H1(G, K*) = 1. The second cohomology group H2(Gal(K/k), K*) is isomorph to Br(K/k). For any field k, Br(k) is a torsion Abelian group.
Let (for V a KG module) K(V) denote the field of fractions of K[V] (the field of rational functions on V). K(V) is a Galois extension (i.e. normal and separable) of K(V)G with Galois group G. K(V)G is the field of fractions of K[V]G, and K[V]G is integrally closed in K(V)G. Dealing with the cohomology of groups H0 are the fixed points of a group, more exactly K[V]G = H0(G, K[V]).
By now all this looks familiar to us, isn't it? All this has to do with algebraic K-theory, field extensions, algebraic number theory and as well arithmetic geometry.