no. 2
Trivialité du 2-rang du noyau hilbertien
Journal de théorie des nombres de Bordeaux, Tome 6 (1994) no. 2, pp. 459-483.

We give exhaustive list of biquadratic fields K=ℚ(i,m) and K=ℚ(2,m) without 2-exotic symbol, i.e. for which the 2-rank of the Hilbert kernel (or wild kernel) is zero. Such K=ℚ(i,m) are logarithmic principals [J3]. We detail an exemple of this technical numerical exploration and quote the family of theories and results we utilize. The 2-rank of tame, regular and wild kernel of K-theory are connected with local and global problem of embedding in a Z 2 -extension. Global class field theory can describe the 2-rank of the Hilbert kernel and reveals existence of symbols on K not given by local class field theory.

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     author = {Thomas, Herv\'e},
     title = {Trivialit\'e du $2$-rang du noyau hilbertien},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {459--483},
     publisher = {Universit\'e Bordeaux I},
     volume = {6},
     number = {2},
     year = {1994},
     mrnumber = {1360655},
     zbl = {0822.11079},
     language = {fr},
     url = {http://archive.numdam.org/item/JTNB_1994__6_2_459_0/}
}
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Thomas, Hervé. Trivialité du $2$-rang du noyau hilbertien. Journal de théorie des nombres de Bordeaux, Tome 6 (1994) no. 2, pp. 459-483. http://archive.numdam.org/item/JTNB_1994__6_2_459_0/

[BT] H. Bass and J. Tate, The Milnor ring of a global field, (with an appendix by J. Tate), in Algebraic K-theory II. Lecture Notes in Mathematics, 342, Springer-Verlag, 1973. Berlin-Heidelberg- New York. | MR | Zbl

[BP] F. Bertrandias et J.-J. Payan, Γ-extensions et invariants cyclotomiques, Ann. scient. Éc. Norm. Sup. 5 (1972), 517-548. | Numdam | Zbl

[BS] J. Browkin and A. Schinzel, On Sylow 2-subgroups of K2OF for quadratic number fields F, J. reine angew. Math. 331 (1982), 104-113. | MR | Zbl

[Br] A. Brumer, On the units of algebraic number field, Mathematika 14 (1967), 121-124. | MR | Zbl

[Co] J. Coates, p-adic L-functions and Iwasawa theory, in Durham symposium in algebraic number field, (A. Frôlich editor), Academic Press, 1977. New York, London. | MR | Zbl

[CH] P.E. Conner and J. Hurrelbrink, A comparison theorem for the 2-rank of K2D, Contemporary Mathematics 55, Part II (1986), 411-420. | MR | Zbl

[Ga] H. Garland, A finiteness theorem for K2 of a number field, Annals of Math. 94 (1971), 534-548. | MR | Zbl

[Gi] R. Gillard, Formulations de la conjecture de Leopoldt et étude d'une condition suffisante, Abh. Math. Sem. Hambourg 48 (1979), 125-138. | MR | Zbl

[G1] G. Gras, Groupe de Galois de la p-extension abélienne p-ramifiée maximale d'un corps de nombres, J. reine angew. Math. 333 (1982), 86-132. | MR | Zbl

[G2] G. Gras, Plongements kummeriens dans les Zp-extensions, Compositio Math. 55 (1985), 383-396. | Numdam | MR | Zbl

[GJ] G. Gras et J.-F. Jaulent, Sur les corps de nombres réguliers, Math. Z. 202 (1989), 343-365. | MR | Zbl

[J1] J.-F. Jaulent, L'arithmétique des l-extensions, (thèse) Pub. Math. Fac. Sci. Besançon, Théor. Nombres 1984-1985 & 1985-1986 (1986), 1-348. | MR | Zbl

[J2] J.-F. Jaulent, Sur les conjectures de Leopoldt et Gross, in Journées arithmétiques de Besançon, Astérisque 147-148 (1987), 107-120. | MR | Zbl

[J3] J.-F. Jaulent, La théorie de Kummer et le K2 des corps de nombres, J. Théor. Nombres Bordeaux 6 (1994).

[J4] J.-F. Jaulent, Sur le noyau sauvage des corps de nombres, Acta Arith. 67 (1994), 335-348. | MR | Zbl

[KC] K. Kramer et A. Candiotti, On K2 and Zl- extensions of numbers fields, Amer. J. Math. 100 (1978), 177-196. | MR | Zbl

[Ma] H. Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. scient. Éc. Norm. Sup. 4 2 (1969), 1-62. | Numdam | MR | Zbl

[MN] A. Movahhedi & T. Nguyen Quang Do, Sur l'arithmétique des corps de nombres p-rationnels, Sém. Th. des Nbres Paris (1987/1988), Prog. Math. 102 (1990), 155-197. | MR | Zbl

[Ti] J. Tate, Symbols in arithmetics, Actes Congrès intern. math., Tome 1 (1970), 201-211. | MR | Zbl

[T2] J. Tate, Relation between K2 and Galois cohomology, Invent. Math. 36 (1976), 257-274. | MR | Zbl

[Th] H. Thomas, Premier étage d'une Zl-extension, Manuscripta Math. 81 (1993), 413-435. | MR | Zbl

[Wh] K.S. Williams, Integers of biquadratic fields, Canad. Math. Bull. 13 (1970), 519-528. | MR | Zbl

[Wi] A. Wiles, The Iwasawa conjecture for totally real fields, Annals of Math. 131 (1990), 493-540. | MR | Zbl