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Leopold Kronecker wrote, "All results of the profoundest mathematical investigation must ultimately be expressible in the simple form of properties of integers." It is perhaps this feeling that made him say, "God made the integers, all the rest is work of humanity." Elementary number theory is related to the properties of the natural numbers 1, 2, 3,..n. In the course of studying these properties it become requisite to take into account the ring of integers Z and then the field of rational numbers Q. The first systemic investigation of integers lying in an algebraic number field not necessary Q was carried out by Gauss (Gauss 1832) who considered the integers of the quadratic number field Q(i). In his effort to realize bi-quadratic reciprocity he introduced the ring Z(i). similarly to study higher reciprocity laws and Fermat's Last theorem other rings of integers were unfixed. Finally a general definition of an algebraic number field and ring of algebraic integers came forth, principally through the efforts of E. Kummer and R. Dedekind. Our work is related to the Hasse's problem for the construction of integral bases of Pure sextic number fields and also to check their monogeneity.