László Remete

Integral bases and monogenity of pure fields

Abstract Let m be a square-free integer ( m ≠ 0 , ± 1 ). We show that the structure of the integral bases of the fields K = Q ( m n ) is 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.

Integral Bases and Monogenity of Composite Fields

ABSTRACT We consider infinite parametric families of high degree number fields composed of quadratic fields with pure cubic, pure quartic, pure sextic fields and with the so called simplest cubic, simplest quartic fields. We explicitly describe an integral basis of the composite fields. We construct the index form, describe their factors and prove that the monogenity of the composite fields imply certain divisibility conditions on the parameters involved. These conditions usually cannot hold, which implies the non-monogenity of the fields. The fields that we consider are higher degree number fields, of degrees 6 up to 12. The non-monogenity of the number fields is stated very often as a consequence of the non-existence of the solutions of the index form equation. As per our knowledge, it is not at all feasible to solve the index form equation in these high degree fields, especially not in a parametric form. On the other hand, our method implies directly the non-monogenity in almost all cases. We obtain our results in a parametric form, characterizing these infinite parametric families of composite fields.

Calculating power integral bases by solving relative Thue equations

Abstract In our recent paper I. Gaál: Calculating “small” solutions of relative Thue equations, J. Experiment. Math. (to appear) we gave an efficient algorithm to calculate “small” solutions of relative Thue equations (where “small” means an upper bound of type 10500 for the sizes of solutions). Here we apply this algorithm to calculating power integral bases in sextic fields with an imaginary quadratic subfield and to calculating relative power integral bases in pure quartic extensions of imaginary quadratic fields. In both cases the crucial point of the calculation is the resolution of a relative Thue equation. We produce numerical data that were not known before.

Non-monogenity in a family of octic fields

Let

Calculating power integral bases by using relative power integral bases

m

Binomial Thue equations and power integral bases in pure quartic fields.

be a square-free positive integer,

Simplest quartic and simplest sextic Thue equations over imaginary quadratic fields

m\equiv 2,3 \; (\bmod \; 4)

Totally real Thue inequalities over imaginary quadratic fields

. We show that the number field

Integral bases and monogenity of the simplest sextic fields

K=Q(i,\sqrt[4]{m})

Power integral bases in a family of sextic fields with quadratic subfields

is non-monogene, that is it does not admit any power integral bases of type