Geodesic
Explicit form of the law of reciprocity
Izvestiya. Mathematics , Tome 13 (1979) no. 3, pp. 557-588

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A pairing in the multiplicative group of a local field (a finite extension of a $p$-adic number field) is defined in terms of the expansion of elements into series in a local uniformizing parameter. The main properties of this pairing are proved: bilinearity, skew-symmetry, invariance with respect to choice of local uniformizing parameter, and independence of the choice of expansion of elements into series in this uniformizing parameter. In addition, the norm property of this pairing is examined. The main result of this paper is that this pairing coincides with Hilbert's norm residue symbol. This yields an explicit formula for the latter. Bibliography: 9 titles. 
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     author = {S. V. Vostokov},
     title = {Explicit form of the law of reciprocity},
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S. V. Vostokov. Explicit form of the law of reciprocity. Izvestiya. Mathematics , Tome 13 (1979) no. 3, pp. 557-588. http://geodesic.mathdoc.fr/item/IM2_1979_13_3_a2/