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Let L = Q[a] be a cyclic number field of odd prime degree q over the field Q of rationals. In this paper we give an algorithm to compute the discriminant of £ / Q , which relies upon a fast method to find Eisenstein elements in L. The algorithm accepts as input the minimal polynomial of a over Q and a complete factorisation of the discriminant of a, and computes, in time polynomial in the size of the input, a list consisting of all the ramified primes with corresponding Eisenstein elements.doi:10.1017/s0004972700014763 fatcat:4bwa3dcserbxnnhidc5i6os7w4