Narrow Results

Several relations and bounds for the dimension of principal ideals in group algebras are determined by analyzing minimal polynomials of regular representations. These results are used in the two last sections. First, in the context of semisimple group algebras, to compute, for any abelian code, an element with Hamming weight equal to its dimension. Finally, to get bounds on the minimum distance of certain MDS group codes. A relation between a class of group codes and MDS codes is presented. Examples illustrating the main results are provided.

For a general class of non-negative functions defined on integral ideals of number fields, upper bounds are established for their average over the values of certain principal ideals that are associated to irreducible binary forms with integer coefficients.

In this paper, we study different properties of Stone almost distributive lattice (Stone ADL), prove basic facts on a Stone ADL and derive a necessary and sufficient conditions for an ADL L to be a Stone ADL.

Let K be a number field. The Pólya field concept is used to know when the module of integer-valued polynomials over the ring of integers ${{𝒪}}_K$ of K has a regular basis. In [C. W.-W. Tougma, Some questions on biquadratic Pólya fields, J. Number Theory 229 (2021) 386–398], the author used cohomological results from [C. B. Setzer, Units over totally real $C_2\times C_2$ fields, J. Number Theory 12 (1980) 160–175] to answer questions raised in [A. Leriche, Pólya fields, Pólya groups and Pólya extensions: a question of capitulation, J. Théor. Nr. Bordx. 23 (2011) 235–249; A. Leriche, Cubic, quartic and sextic Pólya fields, J. Number Theory  133 (2013) 59–71] on biquadratic Pólya fields. Here we first prove that number fields were omitted from the list of exceptional fields cited in [A. Leriche, Pólya fields, Pólya groups and Pólya extensions: a question of capitulation, J. Théor. Nr. Bordx. 23 (2011) 235–249; A. Leriche, Cubic, quartic and sextic Pólya fields, J. Number Theory  133 (2013) 59–71]. We therefore identify new biquadratic Pólya fields, where the prime number 2 is totally ramified. This result corrects and completes some others on the literature. On the other hand, we show that the main results of [C. W.-W. Tougma, Some questions on biquadratic Pólya fields, J. Number Theory  229 (2021) 386–398] and this paper can be proved with a single method using Kuroda/Kubota’s unit formula without cohomological results of [C. B. Setzer, Units over totally real $C_2\times C_2$ fields, J. Number Theory  12 (1980) 160–175].