Shafarevich's paper ``A general reciprocity law'' (Q2848649)

Hilbert's problem nine asks to find the most general Reciprocity Law in any number field. Once class field theory had been established, the classical reciprocity law for the product of power residues in an algebraic number field was reduced to calculate the Hilbert's norm-residue symbol in local fields. In this way, the problem of calculating the Hilbert symbol is the final stage in Hilbert's ninth problem.NEWLINENEWLINEThe results of \textit{I. R. Shafarevich} [Mat. Sb., N. Ser. 26(68), 113--146 (1950; Zbl 0036.15901)] played an important role in obtaining explicit formulae for the Hilbert symbol. In this paper, the author uses a new approach to obtain explicit formulae for the reciprocity law based on Witt's idea of identifying the Brauer group of the field with the group of primary elements.NEWLINENEWLINELet \(K\) be a local field (a finite extension of \({\mathbb Q}_p\)) containing the \(p^n\)-th roots of unity \(\langle \zeta\rangle\), then we can define the Hilbert symbol NEWLINE\[NEWLINE (\cdot,\cdot): K^{\ast}\times K^{\ast}\to \langle \zeta \rangle NEWLINE\]NEWLINE given by \((\alpha,\beta)=\root p^n \of {\beta}^{\psi_K(\alpha)-1}\), where \(\psi_K: K^{\ast}\to \mathrm{Gal}(K^{ab}/K)\) is the local reciprocity map. The main result of this paper, Theorem 1, gives an explicit formula for the Hilbert symbol when \(p\neq 2\).NEWLINENEWLINETo prove the theorem, the author first reduces the Hilbert symbol on a pair \((\alpha,\beta)\), \(\alpha,\beta\in K^{\ast}\), to the symbol \((\pi,\varepsilon)\) where \(\varepsilon\) is a principal unit and \(\pi\) is a prime element. Next, he calculates the primary elements and defines explicit generators of the group of principal units. Finally, the Hilbert symbol on the pair \((\pi,\varepsilon)\) is calculated.NEWLINENEWLINEIn the second part of the paper, using the same method, formulae for a complete higher--dimensional field with perfect residue field are obtained.