Représentation de \(-1\) comme somme de carrés d'entiers dans un corps quadratique imaginaire (Q2554730)

Let \(A_m\) denote the ring of integers of the imaginary quadratic field \(\mathbb Q(\sqrt{-m})\), where \(m\) is a square free integer. By elementary methods the author determines the stufe \(s(A_m)\) of \(A_m\), i.e. the least number of squares of integers of \(A_m\) which represent \(-1\). The detailed results are as follows: Theorem 1. If \(m\equiv 7\pmod 8\), then \(s(A_m) = 4\). Theorem 2. If \(m>1\) and \(m\equiv 3\pmod 4\), then \(s(A_m) = 2\) or \(3\) according as norm of \(\varepsilon_m\) is \(-1\) or \(+1\), where \(\varepsilon_m\) is the fundamental unit of \(\mathbb Q(\sqrt{m})\). Theorem 3. If \(m\) is congruent to 3 modulo 8, then the following three statements are equivalent: (i) \(s(A_m) =2\). (ii) The equation \(x^2 - my^2 = -2\) is solvable in integers. (iii) If \(\varepsilon_m = u_m + v_m\sqrt m\), then \(u_m - 1\) is a square in the ring of rational integers. (Here \(\varepsilon_m\) is as in theorem 2.) If these conditions are not satisfied, then \(s(A_m) = 3\). The author proves that \(s(A_m) = 2\), if \(m\) is a prime and \(m\equiv 7\pmod 8\) (Proposition 3).