On the corestriction of \(p^ n\)-symbol
Let \(F\) be a field of characteristic \(p \neq 0\) and let \(W_ n(F)\) denote the group of truncated Witt vectors of length \(n\) over \(F\). A \(p^ n\)- symbol over \(F\) is a central simple \(F\)-algebra of degree \(p^ n\) generated by elements \(\theta_ 1,\ldots, \theta_ n, u\) subject to the relations: \(u^{p^ n}=a\), \(\theta^ p-\theta=\omega\) and \(u\theta u^{-1}= \theta+1\), where \(a \in F^ \times\), \(\omega \in W_ n(F)\) and \(\theta=(\theta_ 1,\ldots, \theta_ n) \in W_ n(F(\theta_ 1,\ldots,\theta_ n))\). According to a result of Witt every cyclic algebra of degree \(p^ n\) over \(F\) is a \(p^ n\)-symbol. On the other hand, it follows from classical work of Albert and Teichmüller that every central simple \(F\)-algebra of exponent \(p^ e\) is Brauer- equivalent to a tensor product of \(p^ e\)-symbols. The purpose of the present paper is to give an upper bound for the number of factors in the decomposition of the Brauer class of certain central simple \(F\)-algebras of index \(p^ n\). If \(L/F\) is a finite separable extension of degree \(r\), it is shown that the corestriction of every \(p^ n\)-symbol over \(L\) is Brauer-equivalent to a tensor product of at most \(r\) \(p^ n\)-symbols over \(F\). An example is given to show that this bound is sharp. Using this result, the authors also show that cyclic \(F\)- algebras of degree \(p^ n\) and exponent \(p^ e\) are Brauer-equivalent to tensor products of \(p^ e\)-symbols with at most \(p^{n-e}\) factors.
scientific article; zbMATH DE number 78194