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On quasi algebraic closure. (English) Zbl 0046.26202
A field $$F$$ is called to be $$C_i$$, if every form in $$F$$ of degree $$d$$ in $$n$$ variables with $$n > d^i$$ has a non trivial zero in $$F$$. A $$C_0$$ field $$(n = d)$$ is algebraically closed and a $$C_1$$ field is quasi algebraically closed (Artin). The author proves:
Let $$F$$ be $$C_i$$ and suppose $$F$$ admits at least one normic form of order $$i$$, which is a form in $$F$$ with $$n = d^i$$ having only the trivial zero in $$F$$. Then any finite extension of $$F$$ is also $$C_i$$. If now $$F$$ be a function field in $$k$$ variables over a $$C_i$$ constant field with a normic form of order $$i$$, then $$F$$ is $$C_{i+k}$$. If $$F$$ be a field complete under a discrete valuation with algebraically closed residue class field, then $$F$$ is $$C_1$$, from which is deduced that some fields are really $$C_1$$, for example: The maximal unramified extension of a field complete under a discrete valuation with perfect residue class field is $$C_1$$ (Artin’s conjecture).
Applying these results to class field theory, the author proves:
Let $$F$$ by any field and let $$\Omega$$ be an extension of $$F$$ which is $$C_1$$. Then every cocycle is split by a finite subfield of $$\Omega$$. If now $$F$$ be complete under a discrete valuation with finite residue class field, then $$H^2(F)$$ (the second cohomology group) is isomorphic with the rationals (mod 1). Then any cocycle of exponent $$n$$ is split by any field of degree $$n$$ over $$F$$ (Chevalley).
In case of function fields it is proved:
If $$F$$ be a function field of one variable over a constant field, then every cocycle has a splitting field which is a finite extension of the constant field.
It seems not easy, to extend local arithmetic results to a number field in the large.
Reviewer: Z. Suetuna (Tokyo)
Show Scanned Page ##### MSC:
 11Rxx Algebraic number theory: global fields 11R58 Arithmetic theory of algebraic function fields 11R34 Galois cohomology
##### Keywords:
number fields; function fields
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