On quasi algebraic closure.

*(English)*Zbl 0046.26202A 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.

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)