An octic reciprocity law of Scholz type

Abstract:

The authors [3] have conjectured that if p and q are distinct primes satisfying \[ p \equiv q \equiv 1\quad \pmod 8,\quad {(p/q)_4} = {(q/p)_4} = + 1,\] then \[ {\left ( {\frac {p}{q}} \right )_8}{\left ( {\frac {q}{p}} \right )_8} = \left \{ {\begin {array}{*{20}{c}} {{{\left ( {\frac {{{\varepsilon _p}}}{q}} \right )}_4}{{\left ( {\frac {{{\varepsilon _q}}}{p}} \right )}_4},\quad {\text {if}}\;N({\varepsilon _{pq}}) = - 1,} \hfill \\ {{{( - 1)}^{h(pq)/4}}{{\left ( {\frac {{{\varepsilon _p}}}{q}} \right )}_4}{{\left ( {\frac {{{\varepsilon _q}}}{p}} \right )}_4},\quad {\text {if}}\;N({\varepsilon _{pq}}) = + 1,} \hfill \\ \end {array} } \right .\] where ${\varepsilon _p}$ is the fundamental unit of $Q(\sqrt p ),N({\varepsilon _{pq}})$ denotes the norm of the unit ${\varepsilon _{pq}}$, and $h(pq)$ is the class number of $Q(\sqrt {pq} )$. A proof of this conjecture is given, which makes use of results of Bucher [2].