# Factor system

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In mathematics, a **factor system** is a function on a group giving the data required to construct an algebra. A factor system constitutes a realisation of the cocycles in the second cohomology group in group cohomology.

Let *G* be a group and *L* a field on which *G* acts as automorphisms. A *cocycle* or *factor system* is a map *c*:*G* × *G* → *L*^{*} satisfying

Cocycles *c* and *c'*are *equivalent* if there exists some system of elements *a* : *G* → *L*^{*} with

Cocycles of the form

are called *split*. Cocycles under multiplication modulo split cocycles form a group, the second cohomology group H^{2}(*G*,*L*^{*}).

## Crossed product algebras

Let us take the case that *G* is the Galois group of a field extension *L*/*K*. A factor system in H^{2}(*G*,*L*^{*}) gives rise to a *crossed product algebra* *A*, which is a *K*-algebra containing *L* as a subfield, generated by the elements λ in *L* and *u*_{g} with multiplication

Equivalent factor systems correspond to a change of basis in *A* over *K*. We may write

Every central simple algebra over *K* that splits over *L* arises in this way. The tensor product of algebras corresponds to multiplication of the corresponding elements in H^{2}. We thus obtain an identification of the Brauer group, where the elements are classes of CSAs over *K*, with H^{2}.^{[1]}

## Cyclic algebra

Let us further restrict to the case that *L*/*K* is cyclic with Galois group *G* of order *n* generated by *t*. Let *A* be a crossed product (*L*,*G*,*c*) with factor set *c*. Let *u* = *u*_{t} be the generator in *A* corresponding to *t*. We can define the other generators

and then we have *u*^{n} = *a* in *K*. This element *a* specifies a cocycle *c* by

It thus makes sense to denote *A* simply by (*L*,*t*,*a*). However *a* is not uniquely specified by *A* since we can multiply *u* by any element λ of *L*^{*} and then *a* is multiplied by the product of the conjugates of λ. Hence *A* corresponds to an element of the norm residue group *K*^{*}/N_{L/K}*L*^{*}. We obtain the isomorphisms

## References

- ↑ Saltman (1999) p.44