This paper presents an effective description of the universal design cubic cyclic fields and using an algorithm to compute the minimum polynomials generating elements. The calculations are carried out with the use of the properties of Gauss sums corresponding primitive Dirichlet character. Built fundamental and integral basis normal cubic cyclic fields. Necessary and sufficient criterion of element reversibility of totally solid cyclic fields is given here. The different examples of

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... l exercises of minimal polynomials have been mentioned here. Constructing the model Abelian field, Kronecker-Weber theorem was taken into account; it states that any Abelian extension of the field of rational numbers is in some cyclotomic field. This theorem gives a classification of Abelian fields, and also determines the laws of decomposition, makes it possible to determine the structure of their discriminants and to obtain explicit expressions for the number of ideal classes. The results of the work describe Abelian extensions of a given field. Obtained theoretical data may be used in subsequent studies of algebraic theories, namely, the problem of absolutely Abelian fields.