Dans cet article nous appliquons le foncteur corps des normes pour déduire, dans le cas de caractéristique mixte, une formule explicite pour le symbole de Hilbert de la formule explicite pour le symbole de Witt en caractéristique dans le contexte des corps locaux multidimensionnels. On montre que la formule explicite de Vostokov est un cas très particulier de notre construction.
The field-of-norms functor is applied to deduce an explicit formula for the Hilbert symbol in the mixed characteristic case from the explicit formula for the Witt symbol in characteristic in the context of higher local fields. Is is shown that a “very special case” of this construction gives Vostokov’s explicit formula.
Mots clés : higher local fields, field-of-norms, Hilbert Symbol, Vostokov’s pairing
@article{JTNB_2012__24_1_1_0,
author = {Abrashkin, Victor and Jenni, Ruth},
title = {The field-of-norms functor and the {Hilbert} symbol for higher local fields},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {1--39},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {24},
number = {1},
year = {2012},
doi = {10.5802/jtnb.786},
mrnumber = {2914899},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jtnb.786/}
}
TY - JOUR AU - Abrashkin, Victor AU - Jenni, Ruth TI - The field-of-norms functor and the Hilbert symbol for higher local fields JO - Journal de théorie des nombres de Bordeaux PY - 2012 SP - 1 EP - 39 VL - 24 IS - 1 PB - Société Arithmétique de Bordeaux UR - http://www.numdam.org/articles/10.5802/jtnb.786/ DO - 10.5802/jtnb.786 LA - en ID - JTNB_2012__24_1_1_0 ER -
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Abrashkin, Victor; Jenni, Ruth. The field-of-norms functor and the Hilbert symbol for higher local fields. Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 1, pp. 1-39. doi : 10.5802/jtnb.786. http://www.numdam.org/articles/10.5802/jtnb.786/
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