Abstract
We present an algorithm for the computation of the discrete logarithm in logarithmic l-Class Groups. This is applied to the calculation of the l-rank of the wild kernel WK 2(F) of a number field F. In certain cases it can also be used to determine generators of the l-part of WK 2(F).
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References
Banaszak, G.: Generalization of the Moore exact sequence and the wild kernel for higher K-groups. Compos. Math. 86(3), 281–305 (1993)
Bass, H., Tate, J.: The Milnor ring of a global field. In: Bass, H. (ed.) Algebraic K-theory. Lecture Notes in Math., vol. 342, pp. 349–446. Springer, Berlin (1973)
Belabas, K., Gangl, H.: Generators and Relations for K\(_2{\mathcal{O}}_F\) (preprint)
Canon, J.J., et al.: The computer algebra system Magma, University of Sydney (2003), http://magma.maths.usyd.edu.au/magma/
Diaz y Diaz, F., Jaulent, J.-F., Pauli, S., Pohst, M.E., Soriano, F.: A new Algorithm for the Computation of logarithmic l-class groups of number fields. submitted to Experimental Mathematics
Diaz y Diaz, F., Soriano, F.: Approche algorithmique du groupe des classes logarithmiques. J. Number Theory 76, 1–15 (1999)
Federer, L.J., Gross, B.H. (with an appendix by W. Sinnot) Regulators and Iwasawa modules. Invent. Math. 62, 443–457 (1981)
Garland, H.: A finiteness theorem for K2 of a number field. Ann. of Math. 94, 534–548 (1971)
Gras, G.: Class Field Theory. Springer Monographs in Mathematics (2003)
Jaulent, J.-F.: Introduction au K2 des corps de nombres. Publ. Math. Fac. Sci. Besançon (Théorie des nombres), Anèes (1984/1985-1985/1986)
Jaulent, J.-F.: Sur le noyau sauvage des corps de nombres. Acta Arithmetica 67, 335–348 (1994)
Jaulent, J.-F.: Classes logarithmiques des corps de nombres. J. Théor. Nombres Bordeaux 6, 301–325 (1994)
Jaulent, J.-F.: Classes logarithmiques des corps de nombres totalement réels. Acta Arithmetica 103, 1–7 (2002)
Jaulent, J.-F., Soriano-Gafiuk, F.: Sur le noyau sauvage des corps de nombres et le groupe des classes logarithmiques. Math. Z. 238, 335–354 (2001)
Jaulent, J.-F., Soriano-Gafiuk, F.: 2-Groupe des classes positives et noyau sauvage de la K2-Théorie, to appear in J. Number Theory
Keune, F.: On the structure of the K2 of the ring of integers in a number field. K-Theory 2 (5), 625–645 (1989); MSC 2000
Kolster, M.: An idelic Approach to the Wild Kernel. Invent. Math. 103, 9–24 (1991)
Milnor, J.: Introduction to Algbraic K-Theory. Ann. of Math. Studies, 72. Princeton Univ. Press, Princeton (1971)
Moore, C.: Group extensions of p-adic and adelic linear groups. Publ. Math. I.H.E.S. 35, 5–74 (1969)
Soriano-Gafiuk, F.: Sur le noyau hilbertien d’un corps de nombres. C. R. Acad. Sci. Paris, t. 330, Série I, 863–866 (2000)
Tate, J.: Symbols in arithmetic. Actes Congrès Int. de Math., Tome 1, 201–211 (1970)
Tate, J.: Relations between K2 and Galois cohomology. Invent. Math. 36, 257–274 (1976)
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Pauli, S., Soriano-Gafiuk, F. (2004). The Discrete Logarithm in Logarithmic l-Class Groups and Its Applications in K-theory. In: Buell, D. (eds) Algorithmic Number Theory. ANTS 2004. Lecture Notes in Computer Science, vol 3076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24847-7_28
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DOI: https://doi.org/10.1007/978-3-540-24847-7_28
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