Geodesic
Hilbert symbol in a discrete valuated field
Zapiski Nauchnykh Seminarov POMI, Rings and modules. Part 2, Tome 94 (1979), pp. 50-69

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In a complete discrete valuated field of characteristic zero with perfect residue field of prime characteristic a pairing with values in the group of primary elements is defined. It is proved that this pairing agrees with the Hilbert norm residue symbol and therefore provides an explicit formula for the latter. 
@article{ZNSL_1979_94_a5,
     author = {S. V. Vostokov},
     title = {Hilbert symbol in a discrete valuated field},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {50--69},
     publisher = {mathdoc},
     volume = {94},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_94_a5/}
}
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S. V. Vostokov. Hilbert symbol in a discrete valuated field. Zapiski Nauchnykh Seminarov POMI, Rings and modules. Part 2, Tome 94 (1979), pp. 50-69. http://geodesic.mathdoc.fr/item/ZNSL_1979_94_a5/