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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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##### References:
 [1] Artin, M, Dimension cohomologique: premiers résultats, (), 43-63, Tome 3 · Zbl 0269.14007 [2] Ax, J, A field of cohomological dimension 1 which is not C1, Bull. amer. math. soc., 71, 717, (1965) · Zbl 0142.29902 [3] Baeza, R, (), Lecture Notes in Mathematics [4] Bass, H; Tate, J, The Milnor ring of a global field, (), 349-446 · Zbl 0299.12013 [5] Bourbaki, N, Éléments de mathématique: algèbres, (1959), Hermann Paris, Chap. 9 · Zbl 0102.25503 [6] Bourbaki, N, Éléments de mathématique: espaces vectoriels topologiques, (1953), Herman Paris, Chap. 1 · Zbl 0050.10703 [7] Elman, R; Lam, T.Y, Pfister forms and K-theory of fields, J. algebra, 23, 181-213, (1972) · Zbl 0246.15029 [8] Greenberg, M.J, Rational points in Henselian discrete valuation rings, Publ. math. IHES, 31, 59-64, (1966) · Zbl 0142.00901 [9] Kato, K, A generalization of local class field theory by using K-groups, I, J. fac. sci. univ. Tokyo sec. IA math., 26, 303-376, (1979) · Zbl 0428.12013 [10] Kato, K, A generalization of local class field theory by using K-groups, II, J. fac. sci. univ. Tokyo sec. IA math., 27, 602-683, (1980) · Zbl 0463.12006 [11] Kato, K, Galois cohomology of complete discrete valuation fields, (), 215-238 [12] Kato, K; Saito, S, Unramified class field theory of arithmetical surfaces, Ann. of math., 118, 241-275, (1983) · Zbl 0562.14011 [13] Lam, T.Y, () [14] Lang, S, On quasi algebraic closure, Ann. of math., 55, 373-390, (1952) · Zbl 0046.26202 [15] Merkuriev, A.S; Suslin, A.A, K-cohomology of Severi-Brauer variety and norm residue homomorphism, Math. USSR-izv., 21, 307-340, (1983) · Zbl 0525.18008 [16] Milne, J.S, Duality in flat cohomology of a surface, Ann. sci. ecole norm. sup., 9, 171-202, (1976) · Zbl 0334.14010 [17] Milnor, J, Algebraic K-theory and quadratic forms, Invent. math., 9, 318-344, (1970) · Zbl 0199.55501 [18] Milnor, J, Introduction to algebraic K-theory, Ann. math. stud., 72, (1971) · Zbl 0237.18005 [19] Pfister, A, Quadratische formen in beliebigen Körpen, Invent. math., 1, 116-132, (1966) · Zbl 0142.27203 [20] Serre, J.-P, (), Lecture Notes in Mathematics [21] Terjanian, G, Un contre-example à une conjecture d’Artin, C. R. acad. sci. Paris, 262, 612, (1966) · Zbl 0133.29705
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