The quadratic and quartic character of certain quadratic units. I (Q1229254)

Let \(\varepsilon_n\) denote the fundamental unit of the real quadratic field \(\mathbb Q(\sqrt m)\). Interpreting \(\sqrt m\) as a rational integer modulo \(p\), the authors study the relationship between (rational) quadratic and biquadratic residue characters of \(\varepsilon_n\) modulo \(p\), and representations of \(p\) by positive-definite binary quadratic forms. Following an idea of \textit{P. Barrucand} and \textit{H. Cohn} [J. Reine Angew. Math. 238, 67--70 (1969; Zbl 0207.36202)], they obtain results related to the 21 imaginary bicyclic biquadratic fields which have class number 1 and which contain one of \(\sqrt{-1}\), \(\sqrt 2\) or if \(\sqrt{-2}\). In addition to giving new proofs of several known results, the authors prove some conjectures of \textit{Emma Lehmer} [J. Reine Angew. Math. 268/269, 294--301 (1974; Zbl 0289.12007)] and some additional new results. Further results obtained by similar methods are the subject of a second paper with the same title [Rocky Mt. J. Math. 9, 683--692 (1979; Zbl 0386.12002)].