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Remarks on \(K_ 2\) of number fields. (English) Zbl 0589.12010

Let k be a number field with ring of integers R and let \(H^ 0_ 2(k)\) be the kernel of the homomorphism \(K_ 2(R)\to \oplus_{v\quad real}\mu_ 2\) given by the Hilbert symbols at the real places. The author studies the triviality of the p-part of \(H^ 0_ 2(k)\) for Galois p-extensions of number fields. In particular for \(p=2\) or 3 a complete list of abelian p-extensions of \({\mathbb{Q}}\) with trivial p-part of \(H^ 0_ 2(k)\) is given.
Reviewer: M.Kolster

MSC:

11R70 \(K\)-theory of global fields
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
11R42 Zeta functions and \(L\)-functions of number fields
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References:

[1] \( \textsc{J. Browkin}K_2\); \( \textsc{J. Browkin}K_2\) · Zbl 0507.18004
[2] Browkin, J.; Schinzel, A., On sylow 2-subgroups of \(K_2O_F\) for quadratic number fields \(F\), J. Reine Angew. Math., 331, 104-113 (1982) · Zbl 0493.12013
[3] Chase, S. U.; Waterhouse, W. C., Moore’s theorem on uniqueness of reciprocity lows, Invent. Math., 16, 267-270 (1972) · Zbl 0245.12009
[4] Garland, H., A finiteness theorem for \(K_2\) of a number field, Ann. of Math., 94, 534-548 (1971) · Zbl 0247.12103
[5] Gras, G., Groupe de Galois de la \(p\)-extension abélienne \(p\)-ramifiée maximale d’un corps de nombers, J. Reine Angew. Math., 333, 86-132 (1982) · Zbl 0477.12009
[6] Gras, G., Logarithme \(p\)-adique et groupes de Galois, J. Reine Angew. Math., 343, 64-80 (1983) · Zbl 0501.12015
[7] Gras, G., Canonical divisibilities of values of \(p\)-adic \(L\)-functions, (Journées arithmétiques (1980)), Exeter · Zbl 0494.12006
[8] Gras, G., Critère de parité du nombre de classes des extensions abéliennes réelles de \(Q\) de degré impair, Bull. Soc. Math. France, 103, 177-190 (1975) · Zbl 0312.12013
[9] Hurrelbrink, J., \(K_2(0)\) for two totally real fields of degree three and four, (Lecture Notes in Math., Vol. 966 (1982), Springer: Springer Berlin) · Zbl 0502.12011
[10] \( \textsc{J.-F. Jaulent}K_2\); \( \textsc{J.-F. Jaulent}K_2\)
[11] Jaulent, J.-F, Sur quelques représentations \(l\)-adiques liées aux symboles et à la \(l\)-ramification, (Sém. théorie des nombres. Sém. théorie des nombres, Bordeaux (1983-1984)) · Zbl 0545.12006
[12] Jaulent, J.-F, \(S\)-classes infinitésimales d’un corps de nombres algébriques, Ann. Sci. Inst. Fourier, 34, 2 (1984) · Zbl 0522.12014
[13] Jaulent, J.-F, Représentations \(l\)-adiques et invariants cyclotomiques (1983-1984), Publ. Math. Fac. Sci: Publ. Math. Fac. Sci Besançon · Zbl 0567.12008
[14] Quillen, D., Higher algebraic \(K\)-theory, algebraic \(K\)-theory I, (Lecture Notes in Math., Vol. 341 (1973), Springer-Verlag: Springer-Verlag Berlin), 85-147 · Zbl 0292.18004
[15] Soulé, C., \(K\)-théorie des anneaux d’entiers de corps de nombres et cohomologie étale, Invent. Math., 55, 251-295 (1979) · Zbl 0437.12008
[16] Tate, J., Relations between \(K_2\) and Galois cohomology, Invent. Math., 36, 257-274 (1976) · Zbl 0359.12011
[17] Tate, J., (Symbols in Arithmetic, Vol. 1 (1970)), 201-211, Actes, Congrès Int. Math.
[18] Urbanowicz, J., On the 2-primary part of a conjecture of Birch and Tate, Acta Arith., 43, 69-81 (1983) · Zbl 0529.12008
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