Supplements to the theory of quartic residues (Q2717607)

Let \(p \equiv 1 \bmod 4\) be prime, \(q\) an odd prime such that \((p/q) = +1\), and put \(q^* = (-1)^{(q-1)/2}q\). The extension \(k(\root 4\of{q^*}\,)\) of \(k = \mathbb Q(\sqrt{-1}\,)\) is the quartic subextension of the ring class field of \(k\) modulo \(4q\), and class field theory implies that the quartic residue symbol \((q^*/p)_4 = 1\) if and only if \(p^{h/4} = x^2 + 16q^2y^2\) for integers \(x, y\), where \(h = q - (-1/q)\) is the ring class number of \(k\) modulo \(4q\). In this paper, the result above is described using the language of quadratic forms, and the author gives an elementary proof.