Binäre quadratische Formen und rationale Zerlegungsgesetze. I. (Binary quadratic forms and rational laws of decomposition. I) (Q1070282)

The author is concerned with the representation of a prime p by a quadratic form on the basis of power residuacity of an algebraic number. For instance \textit{P. Barrucand} and the reviewer [J. Reine Angew. Math. 238, 67-70 (1969; Zbl 0207.362)] showed that if \(p\equiv 1 mod 8\) then \(p=x^ 2+32y^ 2\) exactly when \(1+\sqrt{2}\) is represented rationally modulo p by a quadratic residue. This connection (between forms and residuality) is made through the ring class field (really the class group), which is calculated here in elaborate detail. A general result: Let q (prime) \(\equiv 5 mod 8\) and \(\epsilon\) be a fundamental unit of \({\mathbb{Q}}(\sqrt{q})\), and h the class of number of \({\mathbb{Q}}(\sqrt{q},i)\). Let p (prime) \(\equiv 1 mod 8\), \((q/p)=1\), then \(\epsilon\) is a biquadratic residue of p exactly when \(p^ h=x^ 2+16y^ 2=u^ 2+16qv^ 2\), \(y\equiv v mod 2\) (all in \({\mathbb{Z}})\). This generalizes a conjecture of \textit{E. Lehmer} [ibid. 268/269, 294-301 (1974; Zbl 0289.12007)]. A further result involves eight power residuality for \(\epsilon\) from \({\mathbb{Q}}(\sqrt{q^*})\) for \(q^*\equiv 3 mod 8\).