Sums of three integer squares in complex quadratic fields

Abstract

We classify all complex quadratic number fields that have all their algebraic integers expressible as a sum of three integer squares. These fields are F = Q ( − D ) F = {\mathbf {Q}}(\sqrt { - D} ) , D D a positive square-free integer congruent to 3 ( mod 8 ) 3(\mod 8) and such that D D does not admit a positive proper factorization D ≡ d 1 d 2 D \equiv {d_1}{d_2} that satisfies simultaneously: d 1 ≡ 5 , 7 ( mod 8 ) {d_1} \equiv 5,7(\mod 8) and ( d 2 / d 1 ) = 1 ({d_2}/{d_1}) = 1 .