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**Zyklische Körper und Algebren der Charakteristik \(p\) vom Grad \(p^n\). Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik \(p\).**
*(German)*
Zbl 0016.05101

Consider vectors with denumerably infinitely many components \(x_i\), and let \(p\) be a fixed prime. Then the author defines the sum, difference and product of vectors \(x, y\) such that the components of the resulting vectors are certain polynomials in the components \(x_i\), \(y_j\) with coefficients in the Galois field of \(p\) elements. In particular let the components be in a perfect field \(\mathbf F\) of characteristic \(p\). Then the set \(\mathbf I(\mathbf F)\) of all vectors is an integral domain whose quotient field \(\mathbf Q(\mathbf F)\) is a field of characteristic zero which is complete with respect to a discrete valuation. Moreover \((p)\) is a prime ideal in \(\mathbf Q(\mathbf F)\), that is \(\mathbf Q(\mathbf F)\) is unramified, and the residue-class field of \(\mathbf Q(\mathbf F)\) modulo \((p)\) is equivalent to \(\mathbf F\). Conversely let \(k\) be a field with a discrete valuation and a perfect residue-class field \(\mathbf F\) of characteristic \(p\). Then if \(k\) has characteristic \(p\) it is a power series field in a variable \(t\) over \(\mathbf F\). But when \(k\) has characteristic zero and is complete with respect to the discrete valuation then \(k\) has an invariantively determined subfield \(k'\) equivalent to the \(\mathbf Q(\mathbf F)\) above. Also \((p) = \pi^e\) with \(\pi\) a prime element of \(k\), \(k = k'(\pi)\).

The author next considers vectors of length \(n\) and expresses generations of cyclic fields of degree \(p^n\) over \(k\) of characteristic \(p\) in terms of such vectors. He also defines cyclic algebras \((\alpha\mid \beta]\), where \(\alpha\neq 0\) is in \(k\) and \(\beta\) is a vector defining a cyclic field of degree \(p^n\). Known results on direct products of such cyclic algebras are then expressed in terms of this symbolism.

Finally let \(\mathbf C\) be a perfect field of characteristic \(p\), \(k\) be the power series field in \(t\) over \(\mathbf C\) and let \(\alpha\), \(\beta\) be as above. Then the author defines a residue vector \((\alpha,\beta)\) and proves the residue formula \((\alpha\mid \beta]\) is similar to \((t| (\alpha,\beta)]\). This result is used to obtain invariants of cyclic algebras of degree \(p^n\) over the power series fields \(k\).

The author next considers vectors of length \(n\) and expresses generations of cyclic fields of degree \(p^n\) over \(k\) of characteristic \(p\) in terms of such vectors. He also defines cyclic algebras \((\alpha\mid \beta]\), where \(\alpha\neq 0\) is in \(k\) and \(\beta\) is a vector defining a cyclic field of degree \(p^n\). Known results on direct products of such cyclic algebras are then expressed in terms of this symbolism.

Finally let \(\mathbf C\) be a perfect field of characteristic \(p\), \(k\) be the power series field in \(t\) over \(\mathbf C\) and let \(\alpha\), \(\beta\) be as above. Then the author defines a residue vector \((\alpha,\beta)\) and proves the residue formula \((\alpha\mid \beta]\) is similar to \((t| (\alpha,\beta)]\). This result is used to obtain invariants of cyclic algebras of degree \(p^n\) over the power series fields \(k\).

Reviewer: A. A. Albert (Chicago)