## 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

### Keywords:

cyclic p-algebras

Zbl 0016.05101
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