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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

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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