## 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 $$k$$.

### MSC:

 12J10 Valued fields 11R20 Other abelian and metabelian extensions
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