Hilbert symbol in a complete multidimensional field for an arbitrary prime number. Part I

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.