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A polynomial approach to cocycles over elementary Abelian groups. (English) Zbl 1193.20064
Any map \(\text{GF}(q)\times\text{GF}(q)\to\text{GF}(q)\) may be represented uniquely as a polynomial in \(\text{GF}(q)[x,y]\) of \(x\)-degree and \(y\)-degree at most \(q-1\); here \(q=p^n\), \(p\) a prime. The authors apply this to maps \(\mathbb{Z}_p^n\times\mathbb{Z}_p^n\to\mathbb{Z}_p^n\). They determine which such polynomials correspond to cocycles and which correspond to coboundaries. When \(p=2\), the case of most applications, bases are found for cocyles and coboundaries. A consequence is that each cocycle has a unique representation as a sum of a coboundary and a multiplicative cocycle of a certain type.

MSC:
20J06 Cohomology of groups
11T06 Polynomials over finite fields
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