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The dimension of fields and algebraic K-theory. (English) Zbl 0608.12029
Recall that a field k is said to be $$C_ i$$ if any from of degree d in $$k[x_ 1,...,x_ n]$$ has a non-trivial zero, provided that $$n>d^ i$$. In this article the authors define, for every prime p, a ”p-dimension of k”, $$\dim_ p k$$, which, for p different from the characteristic of k, is the cohomological p-dimension of the profinite group $$gal(k_ s/k)$$, $$k_ s$$ the separable closure of k. They put dim k$$=\sup \dim_ p k$$ and introduce a condition $$C^ q_ i$$ which is similar to $$C_ i$$ but ought to have a stronger relation to the condition dim $$k\leq i.$$
The definition of $$C^ q_ i$$ is as follows. Let $$K_ q(k)$$ be the q- th group of Milnor’s K-theory. For any homogeneous form f over k let $$N_ q(f/k)$$ be the subgroup of $$K_ q(k)$$ generated by the images of the norm maps $$N_{k'/k}: K_ q(k')\to K_ q(k)$$, where k’ runs over the finite extensions of k in which f has non-trivial zeroes. Then k satisfies $$C^ q_ i$$ if, for any finite extension k’ of k and any form f of degree d in $$n>d^ i$$ variables, $$N_ q(f/k')=K_ q(k')$$. It is clear that a $$C_ i$$-field satisfies $$C^ q_ i$$ and it can be proved that a field k satisfies $$C^ 1_ 0$$ if and only if dim $$k\leq 1.$$
The authors conjecture that $$C^ q_ i$$ is equivalent to dim $$k\leq q+i$$. The paper contains several results which support this attractive conjecture. For example, they prove that if k is of positive characteristic p, $$C^ q_ i$$ implies $$\dim_ p k\leq q+i$$.
Reviewer: M.Ojanguren

##### MSC:
 12G10 Cohomological dimension of fields 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 12G05 Galois cohomology 11S70 $$K$$-theory of local fields 11R70 $$K$$-theory of global fields
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