A generalization of Scholz' reciprocity law (Q933160)

Let \(p\) and \(q\) denote two distinct primes \(\equiv 1 \bmod 4\) that are quadratic residues modulo each other, and \(\varepsilon_p\) and \(\varepsilon_q\) the fundamental units of the quadratic number fields with discriminants \(p\) and \(q\), respectively. Scholz's reciprocity law, first proved by \textit{Th. Schönemann} [J. Reine Angew. Math. 19, 289--308 (1839; ERAM 019.0620cj)] in 1839 and then rediscovered in a class field theoretical context by \textit{Arnold Scholz} [Math. Z. 39, 95--111 (1934; Zbl 0009.29402)] in the early 1930s, states that \((\varepsilon_p/q) = (\varepsilon_q/p) = (p/q)_4 (q/p)_4\); one possible way of defining the symbol \((\varepsilon_p/q)\) is through the observation that \(\sqrt{p}\) is a rational integer modulo \(q\) by assumption. \textit{D. A. Buell} and \textit{K. S. Williams} [Am. Math. Mon. 85, 483--484 (1978; Zbl 0383.10004); Proc. Am. Math. Soc. 77, 315--318 (1979; Zbl 0417.10002)] have stated and proved an octic analogue of Scholz's reciprocity law. In this article, the authors follow the reviewer's approach in [Acta Arith. 67, No. 4, 387--390 (1994; Zbl 0833.11049)] for deriving a similar theorem expressing, under suitable assumptions on \(p\) and \(q\), the product \((p/q)_{2^n} (q/p)_{2^n}\) as a power residue symbol of a certain, explicitly given, cyclotomic unit from the subfield of degree \(2^n\) inside the field of \(p\)-th roots of unity.