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An approach to Julia Robinson numbers through the lattice of subfields

Xavier Vidaux, Carlos R. Videla

Abstract

By fully describing the lattice of subfields of some towers of number fields built by iterating square roots, we obtain infinitely many fields, each of them either contradicts Julia Robinson's problem (obtaining a JR-number $4$ which is not a minimum) or gives a Julia Robinson number strictly between four and infinity. This improves a previous result by M. Castillo and the same authors.

An approach to Julia Robinson numbers through the lattice of subfields

Abstract

By fully describing the lattice of subfields of some towers of number fields built by iterating square roots, we obtain infinitely many fields, each of them either contradicts Julia Robinson's problem (obtaining a JR-number which is not a minimum) or gives a Julia Robinson number strictly between four and infinity. This improves a previous result by M. Castillo and the same authors.
Paper Structure (7 sections, 36 theorems, 57 equations, 2 figures)

This paper contains 7 sections, 36 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Thinness for $2$-towers
  3. Notation and some known facts
  4. Quartic extensions within the tower
  5. Towers that are thin from $1$ but not from $0$
  6. Case $\nu=2$
  7. Towers with $\sqrt2$

Key Result

Theorem 1.3

Given $(\nu,x_0)\in\Omega$, distinct from $(2,0)$ and $(2,1)$, the ${\rm JR }$-number of $K^{\nu,x_0}$ either is $4$ and it is not a minimum, or it is strictly between $4$ and $+\infty$.

Figures (2)

  • Figure 1: The lattice of subfields of $K^{\nu,x_0}$ when $u_0-x_0$ is a square and $\nu\ge3$.
  • Figure 2: The lattice of subfields of $K^{2,1}$

Theorems & Definitions (69)

  • Theorem 1.3
  • Theorem 1.4: Determination of the lattice of subfields
  • Corollary 1.5
  • Corollary 1.6
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 59 more