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On Root Capacity, Intersection Indicium, Minimal Generating Sets of Galois Closure & Compositum Feasible Triplets

Shubham Jaiswal

Abstract

We carry on the work started by the author and Bhagwat and develop the theory of root clusters further in this article and also resolve certain problems in related areas. We introduce some new notions as well as recall earlier notions for field extensions over a perfect base field: root cluster size, its generalization root capacity, its dual notion ascending index and its generalization intersection indicium. We establish our results on the Inverse problems for these generalized notions over number fields which generalizes our earlier results. We give a field theoretic formulation for the concept of minimal generating sets of splitting fields which was introduced by the author and Vanchinathan. We present new results as well as generalizations of our earlier results on the cardinalities of minimal generating sets for extensions over number fields. We generalize a result of Drungilas et al. by establishing that a certain family of triplets is compositum feasible over any number field and we also list all the irreducible triplets in this family. We also prove a partial case of a conjecture of Drungilas et al. We also improve on the inverse problems by proving that there exist arbitrarily large finite families of pairwise non-isomorphic extensions having additional properties that satisfy the given conditions. Our methods for all these problems are Galois theoretic in nature and heavily rely on the known cases of the inverse Galois problem.

On Root Capacity, Intersection Indicium, Minimal Generating Sets of Galois Closure & Compositum Feasible Triplets

Abstract

We carry on the work started by the author and Bhagwat and develop the theory of root clusters further in this article and also resolve certain problems in related areas. We introduce some new notions as well as recall earlier notions for field extensions over a perfect base field: root cluster size, its generalization root capacity, its dual notion ascending index and its generalization intersection indicium. We establish our results on the Inverse problems for these generalized notions over number fields which generalizes our earlier results. We give a field theoretic formulation for the concept of minimal generating sets of splitting fields which was introduced by the author and Vanchinathan. We present new results as well as generalizations of our earlier results on the cardinalities of minimal generating sets for extensions over number fields. We generalize a result of Drungilas et al. by establishing that a certain family of triplets is compositum feasible over any number field and we also list all the irreducible triplets in this family. We also prove a partial case of a conjecture of Drungilas et al. We also improve on the inverse problems by proving that there exist arbitrarily large finite families of pairwise non-isomorphic extensions having additional properties that satisfy the given conditions. Our methods for all these problems are Galois theoretic in nature and heavily rely on the known cases of the inverse Galois problem.

Paper Structure

This paper contains 8 sections, 58 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Some Remarks on Unique Intermediate Extensions
  3. Root Capacity and Cluster Towers
  4. Intersection Indicium : A Generalization of Ascending Index
  5. Inverse Root Capacity Problem
  6. Inverse Intersection Indicium Problem
  7. Minimal Generating Sets of Galois Closure
  8. Compositum Feasible Triplets

Key Result

Theorem 1.1

(Inverse Root Capacity Problem for Number Fields) Let $K$ be a number field. Given $(n,r,\rho)$ where $n>2$ and $r|n$ and $r|\rho$ and $\rho\neq n-1$. There exist extensions $L/K$ and $M/K$ such that $[L:K]=n$ and $r_K(L)=r$ and $\rho_K (M,L)=\rho$. For $\rho\neq 0$, we get $M/K$ as an extension of

Theorems & Definitions (128)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.2.1
  • ...and 118 more