arXiv subjects

M Krithika

Publications and source records attributed to M Krithika.

3 recordsLinked to original sources

arXiv2025

On Variants of Inverse Cluster Size Problem & General Magnification

In this article we establish certain variants of the Inverse Cluster Size problem. We introduce the notion of primitive extensions and establish the Primitive variant of the problem. Precisely, we prove the existence of primitive extensions over number fields of any given degree and cluster size less than the degree. We also introduce the notions of Strong and Weak General Magnification and the notion of general primitive extensions. We establish some interesting cases of the General primitive variant of the problem. We also recall the notion of totally real number fields and resolve the Totally real variant of the problem completely.

math.NT↗

arXiv2025

Inflated G-Extensions for Algebraic Number Fields

In 2018, Legrand and Paran proved a weaker form of the Inverse Galois Problem for all Hilbertian fields and all finite groups: that is, there exist possibly non-Galois extensions over given Hilbertian base field with given finite group as the group of field automorphisms fixing the base field. For $\mathbf Q$ it was proved earlier by M. Fried. In this paper our objective is to determine how big the degree of such extension can be compared to the order of the automorphism group. A special case of our result shows that if the Inverse Galois problem for $\bq$ has a solution for a finite group $G$, say of order $n$, then there exist algebraic number fields of degree $nm$, for any $m\ge3$ with the same automorphism group $G$.

math.NT↗

arXiv2023

An Elementary Problem in Galois Theory about the Roots of Irreducible Polynomials

For a field $K$, and a root $\alpha$ of an irreducible polynomial over $K$ (in some algebraic closure) the number of roots of $f(x)$ lying in $K(\alpha)$ is studied here. Given such an $f(x)$ of degree $n$ for which $r$ of the roots are i n $K(\alpha)$, we describe a construction that yields, for $d\ge2$, irreducible polynomials of degree $nd$ and with exactly $rd$ of the roots in the field generated by any one root of those polynomials. Our results are valid for all number fields and possibly some more perfect fields. As an application, for $K=Q$ and positive integers $n\ge3,d\ge2$, we provide irreducible polynomials of degree $nd$ with exactly $d$ roots in the field generated by one of the roots. Independently, for $k<n$, we construct irreducible polynomials over the rationals of degree $n!/(n-k)!$ for which the field generated by one root contains exactly $k!$ roots. Many interesting new questions for further research are provided.

math.NT↗