On the representation of−1 as a sum of two squares in an algebraic number field

AbstractIn a recent paper, P. Chowla proved that −1 can be represented as a sum of two squares in Q(e2πin) if the positive integer n is divisible by a positive integer m ≡ 3(mod 8). In this paper we determine necessary and sufficient conditions in order for −1 to be the sum of two squares in any algebraic number field K. In particular, when K = Q(e2πin) the equation −1 = a2 + b2 is solvable if and only if n is divisible either by 4 or by some odd prime p such that the order of 2(mod p) is even. We show that the set E of such primes consists of all primes ≡ ± 3(mod 8) together with a subset F of the primes ≡ 1(mod 8), where the Dirichlet density of F is 524