Power Integral Bases Research Articles

Power Integral Bases in Composits of Number Fields

AbstractIn the present paper we consider the problem of finding power integral bases in number fields which are composits of two subfields with coprime discriminants. Especially, we consider imaginary quadratic extensions of totally real cyclic number fields of prime degree. As an example we solve the index form equation completely in a two parametric family of fields of degree 10 of this type.

Power integral bases in a parametric family of totally real cyclic quintics

We consider the totally real cyclic quintic fields K n = Q ( ϑ n ) K_{n}=\mathbb {Q}(\vartheta _{n}) , generated by a root ϑ n \vartheta _{n} of the polynomial f n ( x ) = x 5 + n 2 x 4 − ( 2 n 3 + 6 n 2 + 10 n + 10 ) x 3 + ( n 4 + 5 n 3 + 11 n 2 + 15 n + 5 ) x 2 + ( n 3 + 4 n 2 + 10 n + 10 ) x + 1. \begin{multline*} f_{n}(x)=x^{5}+n^{2}x^{4}-(2n^{3}+6n^{2}+10n+10)x^{3}\ +(n^{4}+5n^{3}+11n^{2}+15n+5)x^{2}+(n^{3}+4n^{2}+10n+10)x+1. \end{multline*} Assuming that m = n 4 + 5 n 3 + 15 n 2 + 25 n + 25 m=n^{4}+5n^{3}+15n^{2}+25n+25 is square free, we compute explicitly an integral basis and a set of fundamental units of K n K_{n} and prove that K n K_{n} has a power integral basis only for n = − 1 , − 2 n=-1,-2 . For n = − 1 , − 2 n=-1,-2 (both values presenting the same field) all generators of power integral bases are computed.

Computing all power integral bases in orders of totally real cyclic sextic number fields

An algorithm is given for determining all power integral bases in orders of totally real cyclic sextic number fields. The orders considered are in most cases the maximal orders of the fields. The corresponding index form equation is reduced to a relative Thue equation of degree 3 over the quadratic subfield and to some inhomogeneous Thue equations of degree 3 over the rationals. At the end of the paper, numerical examples are given.

Determining the Monogeneity of a Quartic Number Field

AbstractThis note presents a method that determines all power integral bases of a quartic number field by solving Thue equations of degrees 3 and 4. To this end, projective representations of the ring of integers by graded complete intersections are studied and a criterion for monogeneity in terms of projective representations is derived.

On the Resolution of Index Form Equations in Dihedral Quartic Number Fields

We describe a new algorithm, basedon sieving procedures, for determining the minimal index and all elements with minimal index in a class of totally real quartic fields with Galois group D 8. It is not universally applicable, but its applicability is easily checked for any particular example, and it is very fast when applicable. We include several tables demonstrating the potential of the method. (A more general approach for quartic fields, described in [Gaal et al.], requires much more computation time for each field.) Finally, we present a family of totally real quartic fields with Galois group D 8 and having minimal index 1 (that is, a power integral basis).

Power integral bases in orders of families of quartic fields

Power integral bases in orders of families of quartic fields

On the resolution of index form equations in biquadratic number fields, II

Combining Baker's effective method with the reduction procedure of Baker and Davenport, we give an algorithm for the complete resolution of index form equations corresponding to totally real cyclic biquadratic number fields. The solutions make it possible to construct all power integral bases of these fields. The method can be modified to be applicable also to other types of decomposable form equations.

Computing all power integral bases of cubic fields

Applying Baker’s effective method and the reduction procedure of Baker and Davenport, we present several lists of solutions of index form equations in (totally real and complex) cubic algebraic number fields. These solutions yield all power integral bases of these fields.