On the symbol length of \(p\)-algebras. (Q2852243)

Let \(F\) be a field and \(A\) be a central simple algebra with center \(F\), of degree \(n\). We say \(A\) is a cyclic algebra if it contains a commutative \(F\)-subalgebra \(K\subset A\), Galois over \(F\) such that \(\mathrm{Gal}(K/F)=\langle\sigma\rangle\cong C_n\). The importance of cyclic algebras is expressed in the following theorems:NEWLINENEWLINE-- Teichmüller: Let \(A\) be a finite-dimensional \(F\)-central simple algebra of exponent \(p^e\) and assume the characteristic of \(F\) is \(p\), such algebras are called \(p\)-algebras. Then \(A\) is Brauer equivalent to the tensor product of cyclic algebras of degree \(p^e\).NEWLINENEWLINE-- Merkurjev and Suslin: Let \(A\) be a finite-dimensional \(F\)-central simple algebra of exponent \(p^e\) and assume \(F\) contains a primitive \(p^e\)-root of unity, in particular the characteristic of \(F\) is zero or prime to \(p\). Then \(A\) is Brauer equivalent to the tensor product of cyclic algebras of degree \(p^e\).NEWLINENEWLINE The above raises the following question known as the symbol length problem, ``What is the minimal number of cyclic algebras needed?'' Assuming the case of Merkurjev and Suslin no general answer is known, however if \(F\) contains an algebraically closed field (or more generally a \(C_m\) field) an explicit upper bound was recently found (see [\textit{E. Matzri}, Trans. Am. Math. Soc. 368, No. 1, 413--427 (2016; Zbl 1339.12002)]). For the case of \(p\)-algebras the most general result was by \textit{O. Teichmüller} [Deutsche Math. 1, 362--388 (1936; Zbl 0014.19901; JFM 62.0101.03)] stating that an algebra of index \(p^r\) and exponent \(p^e\) is Brauer equivalent to the tensor product of \(p^r!(p^r!-1)\) cyclic algebras of degree \(p^e\). For algebras of degree \(p\), Mammone improved this bound to \((p-1)!\) (see [\textit{P. Mammone}, Commun. Algebra 14, 517--529 (1986; Zbl 0598.12023)]).NEWLINENEWLINE The main theorem of this paper drastically improves the above mentioned bound for the \(p\)-algebras case, stating that a \(p\)-algebra, \(A\) of index \(p^n\) and exponent \(p^e\) is Brauer equivalent to the tensor product of \(p^n-1\) cyclic algebras of degree \(p^e\).NEWLINENEWLINE The main idea is to twist the Severi-Brauer variety of \(A\) using the Frobenius automorphism to prove the existence of a purely inseparable splitting field for \(A\) of the form \(F(\root{p^e}\of{a_i};\;i=1,\ldots,p^n-1)\). Then using a theorem of Albert (which is independently proved in the paper) stating that such algebras are indeed Brauer equivalent to the tensor product of \(p^n-1\) cyclic algebras of degree \(p^e\).