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P Vanchinathan

Publications and source records attributed to P Vanchinathan.

9 recordsLinked to original sources
arXiv2026

On Minimal generating sets of splitting field, Cluster towers and Multiple transitivity of Galois groups

A natural generating set for a Galois extension regarded as the splitting field of an irreducible polynomial is introduced and investigated here. Minimal generating sets arising in this context throw many surprises compared to the analogous concept in the context of vector spaces: they can be of different cardinalities. In fact we establish that for a certain family of polynomials over the rationals, we have minimal generating sets of all cardinalities in a certain range and that these are the only possible cardinalities for minimal generating set for such a polynomial. We also study how minimal generating sets behave under multiple transitivity of the Galois group and consequently prove the existence of polynomials with all minimal generating sets of uniformly same cardinality. We also connect minimal generating sets with the concept of root cluster tower of an irreducible polynomial introduced in M Krithika, P Vanchinathan (2024).

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arXiv2026

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.

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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 $G$.

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arXiv2024

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

For a field $K$, and a root $α$ of an irreducible polynomial over $K$ (in some algebraic closure) the number of roots of $f(x)$ lying in $K(α)$ is studied here. Given such an $f(x)$ of degree $n$ for which $r$ of the roots are i n $K(α)$, 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.

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arXiv2023

Involutary pemutations over finite fields given by trinomials and quadrinomials

For all finite fields of $q$ elements where $q\equiv1\pmod4$ we have constructed permutation polynomials which have order 2 as permutations, and have 3 terms, or 4 terms as polynomials. Explicit formulas for their coefficients are given in terms of the primitive elements of the field. We also give polynomials providing involutions with larger number of terms but coefficients will be conveniently only two possible values. Our procedure gives at least $(q-1)/4$ trinomials, and $(q-1)/2$ quadrinomials, all yielding involutions with unique fixed points over a field of order $q$. Equal number of involutions with exactly $(q+1)/2$ fixed-points are provided as quadrinomials.

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arXiv2023

Exceptional Quartics are Ubiquitous

For each real quadratic field we constructively show the existence of infinitely many exceptional quartic number fields containing that quadratic field. On the other hand, another infinite collection of quartic exceptional fields without any quadratic subfields is also provided. Both these families are non-Galois extensions of $\mathbf{Q}$, and their normal closu res have Galois groups $D_4$ and $S_4$ respectively. We also show that an infinite number of these exceptional quartic fields have power integral basis, i.e., monogenic. We also construct large collections of exceptional number fields in all degrees greater than 4.

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arXiv2021

An Infinite, Two-parameter Family of Polynomials with Factorization Similar to $X^m-1$

For a suitable irreducible \textit{base} polynomial $f(x)\in \mathbf{Z}[x]$ of degree $k$, a family of polynomials $F_m(x)$ depending on $f(x)$ is constructed with the properties: (i) there is exactly one irreducible factor $Φ_{d,f}(x)$ for $F_m(x)$ for each divisor $d$ of $m$; (ii) deg $(Φ_{d,f}(x))=φ(d)\cdot\mathrm{deg} (f)$ generalizing the factorization of $x^m-1$ into cyclotomic polynomials; (iii) when the base polynomial $f(x) = x-1$ this $F_m(x)$ coincides with $x^m-1$. As an application, irreducible polynomials of degree 12, 24, 24 are constructed having Galois groups of order matching their degrees and isomorphic to $S_3 \oplus C_2 , S_3 \oplus C_2\oplus C_2$ and $S_3 \oplus C_4$ respectively.

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