Geodesic
Hilbert symbol in a complete multidimensional field for an arbitrary prime number. Part I
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 8, Tome 281 (2001), pp. 5-34
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In the first part of this article we discuss two different definitions of Hilbert symbol and prove their equivalence. The second part is devoted to the detailed consideration of the one-dimensional case for an arbitrary prime number $p$ (odd as well as even). At the end of the article we give the explicit formulas in the general case of a multidimensional local field for the both different and mixed characteristic cases for an arbitrary prime number. 
@article{ZNSL_2001_281_a0,
     author = {T. B. Belyaeva and S. V. Vostokov},
     title = {Hilbert symbol in a complete multidimensional field for an arbitrary prime number. {Part~I}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--34},
     year = {2001},
     volume = {281},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_281_a0/}
}
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T. B. Belyaeva; S. V. Vostokov. Hilbert symbol in a complete multidimensional field for an arbitrary prime number. Part I. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 8, Tome 281 (2001), pp. 5-34. http://geodesic.mathdoc.fr/item/ZNSL_2001_281_a0/