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Zerfallende zyklische \(p\)-Algebren. (English) Zbl 0016.05201

The author proves a generalization of a lemma used by Witt in obtaining a residue formula (cf. the first paper Zbl 0016.05101 of Witt reviewed above). Let \(\mathbf C\) be a perfect subfield of a field \(k\) of characteristic \(p\), \(\alpha\neq 0\) in \(k\), \(\mathbf R\) be the ring of all power series \(c_1\alpha+c_2\alpha^2+\ldots\) with \(c_i\) in \(\mathbf C\). Then if \(\beta\) is a vector of length \(n\) with components in \(\mathbf R\) the cyclic algebra \((\alpha\mid \beta]\)is a total matrix algebra. The author also considers the problem of obtaining all vectors \(\beta\) such that \((\alpha\mid \beta]\sim 1\) for fixed \(\alpha\neq 0\) in an arbitrary \(k\) of characteristic \(p\), and obtains a formula for \(\beta\) in terms of the vector symbolism of Witt.

MSC:

13-XX Commutative algebra

Citations:

Zbl 0016.05101
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