Let m be a square-free integer (m = 0, ±1). We show that the structure of the integral bases of the fields K = Q( n √ m) are periodic in m. For 3 ≤ n ≤ 9 we show that the period length is n 2 . We explicitly describe the integral bases, and for n = 3, 4, 5, 6, 8 we explicitly calculate the index forms of K. This enables us in many cases to characterize the monogenity of these fields. Using the explicit form of the index forms yields a new technic that enables us to derive new results on monogenity and to get several former results as easy consequences. For n = 4, 6, 8 we give an almost complete characterization of the monogenity of pure fields.

Let F (x, y) be an irreducible binary form of degree ≥ 3 with integer coefficients and with real roots. Let M be an imaginary quadratic field, with ring of integers Z M . Let K > 0. We describe an efficient method how to reduce the resolution of the relative Thue inequalitiesto the resolution of absolute Thue inequalities of typeWe illustrate our method with an explicit example.

Let M ⊂ K be number fields. We consider the relation of relative power integral bases of K over M with absolute power integral bases of K over Q. We show how generators of absolute power integral bases can be calculated from generators of relative ones. We apply our ideas in infinite families of octic fields with quadratic subfields.

Let m be an integer, m = −8, −3, 0, 5 such that m 2 + 3m + 9 is square free. Let α be a root of f = x 6 − 2mx 5 − (5m + 15)x 4 − 20x 3 + 5mx 2 + (2m + 6)x + 1.The totally real cyclic fields K = Q(α) are called simplest sextic fields and are well known in the literature.Using a completely new approach we explicitly give an integral basis of K in a parametric form and we show that the structure of this integral basis is periodic in m with period length 36. We prove that K is not monogenic except for a few values of m in which cases we give all generators of power integral bases.

Let m be a square-free positive integer, m ≡ 2, 3 (mod 4). We show that the number field K = Q(i, 4 √ m) is non-monogene, that is it does not admit any power integral bases of type {1, α, . . . , α 7 }. In this infinite parametric family of Galois octic fields we construct an integral basis and show non-monogenity using only congruence considerations.Our method yields a new approach to consider monogenity or to prove non-monogenity in algebraic number fields. It is well applicable in parametric families of number fields. We calculate the index of elements as polynomials depending on the parameter, factor these polynomials and consider systems of congruences according to the factors.