Abstract. Let q = ef +1 be a prime number, ζq a q-th primitive root of 1 and η 0 , . . . , η e−1 the periods of degree e of Q(ζq). Write η 0 η i = e−1 j=0 a i,j η j with a i,j ∈ Z. Several characterizations of the numbers η i and a i,j (or, equivalently, of the cyclotomic numbers (i, j) of order e) are given in terms of systems of equations they satisfy and a condition on the linear independence, over Q, of the η i or on the irreducibility, over Q, of the characteristic polynomial of the matrix [a i,j ] 0≤i,j≤e−1 .Let q be an odd prime number, e and f positive integers such that q = ef + 1, s a primitive root modulo q, ζ q a primitive q-th root of 1 and η 0 , η 1 , . . . , η e−1 the Gaussian periods of degree e in Q(ζ q ) defined byDefine η i+je = η i for 0 ≤ i ≤ e − 1 and j ∈ Z. Then Q(η i ) = Q(η 0 ), for any i, and Q(η 0 ) is the only subfield of Q(ζ q ) of degree e over Q. The set {η 0 , η 1 , . . . , η e−1 } is a normal basis of Q(η 0 )/Q and also an integral basis of Q(η 0 ). Let a i,j , 0 ≤ i, j ≤ e − 1, be the rational integers such thatIn this article we show several characterizations of the periods η i , of the integers a i,j and (equivalently) of the cyclotomic numbers (i, j) related to them (see formula (4)). We state such characterizations in terms of some systems of equations satisfied by those numbers, in addition to a condition on the linear independence, over Q, of the η i , or on the irreducibility, over Q, of the characteristic polynomial of the matrix A = [a i,j ] 0≤i,j≤e−1 . The main result, Theorem 1, characterizes the numbers a i,j as the only integral solutions of a system of linear and quadratic equations (the latter corresponding essentially to the effect of the associative law in the multiplication