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

M Krithika, P Vanchinathan

Abstract

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.

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

Abstract

For a field , and a root of an irreducible polynomial over (in some algebraic closure) the number of roots of lying in is studied here. Given such an of degree for which of the roots are i n , we describe a construction that yields, for , irreducible polynomials of degree and with exactly 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 and positive integers , we provide irreducible polynomials of degree with exactly roots in the field generated by one of the roots. Independently, for , we construct irreducible polynomials over the rationals of degree for which the field generated by one root contains exactly roots. Many interesting new questions for further research are provided.
Paper Structure (9 sections, 10 theorems, 2 equations)

This paper contains 9 sections, 10 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of Main Results
  4. Proof for Cluster Magnification
  5. Proof of Theorem \ref{['application']}
  6. Proof of Theorem \ref{['fixkroots']}
  7. Reformulation
  8. Examples
  9. Further Problems

Key Result

Theorem 1

(Cluster Magnification): Let $f(x)\in K[x]$ be the minimal polynomial of an algebraic element $\alpha$ of degree $n>2$ over $K$ with cluster size $r$ and let $K_f$ be its splitting field. Assume that there is a Galois extension of $K$, say of degree $d$, which is linearly disjoint with $K_f$ over $K

Theorems & Definitions (14)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 1
  • Proposition 1
  • Lemma 1
  • Lemma 2
  • proof
  • ...and 4 more