Let m >= 3 be an integer, zeta(m) is an element of C a primitive mth root of unity, and K-m the cyclotomic field Q(zeta(m)). An explicit description of the integral trace form Tr-Km/Q(x (x) over bar)vertical bar Z[zeta(m)] where (x) over bar is the complex conjugate of x is presented. In the case where m is prime, a procedure for finding the minimum of the form subject to x being a nonzero element of a certain Z- module in Z[zeta(m)] is presented.Fundação para o Desenvolvimento da UNESP (FUNDUNESP)San Diego State Univ, Dept Math &Stat, San Diego, CA 92182 USAUniv Estadual Paulista, Dept Matemat, Sao Jose Do Rio Preto, SP, BrazilUniv Estadual Paulista, Dept Matemat, Bauru, SP, BrazilUniv Fed Ceara, Dept Matemat, Fortaleza, Ceara, BrazilUniv Estadual Paulista, Dept Matemat, Sao Jose Do Rio Preto, SP, BrazilUniv Estadual Paulista, Dept Matemat, Bauru, SP, BrazilFUNDUNESP: CCP 2102/2013