On the quartic character of certain quadratic units and the representation of primes by binary quadratic forms (Q1080469)

For a squarefree rational integer m \(>1\), \(\epsilon_ m\) denotes the fundamental unit \((>1)\) of \({\mathbb{Q}}(\sqrt{m})\). For a rational prime \(p\equiv 1\) (mod 4), (\(\cdot /p)\) is the quadratic and \((\cdot /p)_ 4\) the quartic residue symbol (mod p). In 1979 \textit{P. A. Leonard} and the reviewer [ibid. 9, 683-692 (1979; Zbl 0386.12002)] made a number of conjectures involving \((\epsilon_ m/p)\) or \((\epsilon_ m/p)_ 4\) for certain values of m. In this paper the author proves Leonard and Williams's conjecture 3.6. He uses class field theory to prove that if q and q' are primes with \(q\equiv 3\) (mod 8), q'\(\equiv 7\) (mod 8), \((q'/q)=1\), and p is a prime such that \((-1/p)=(2/p)=(q/p)=(q'/p)=1\), then \[ (\epsilon_{qq'}/p)_ 4=(\epsilon_{2q'}/p)_ 4 (- 1)^{y+d},\quad where\quad p^ s=x^ 2+8qq'y^ 2=c^ 2+8d^ 2, \] and s is the odd part of the class number of \({\mathbb{Q}}(\sqrt{qq'}, \sqrt{- 2})\). All of the conjectures made by Leonard and Williams in their 1979 paper have now been proved.