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Even better sums of squares over quintic and cyclotomic fields

Vítězslav Kala, Pavlo Yatsyna

Abstract

We classify all totally real number fields of degree at most 5 that admit a universal quadratic form with rational integer coefficients; in fact, there are none over the previously unsolved cases of quartic and quintic fields. This fully settles the lifting problem for universal forms in degrees at most 5. The main tool behind the proof is a computationally intense classification of fields in which every multiple of 2 is the sum of squares. We further extend these results to some real cyclotomic fields of large degrees and prove Kitaoka's conjecture for them.

Even better sums of squares over quintic and cyclotomic fields

Abstract

We classify all totally real number fields of degree at most 5 that admit a universal quadratic form with rational integer coefficients; in fact, there are none over the previously unsolved cases of quartic and quintic fields. This fully settles the lifting problem for universal forms in degrees at most 5. The main tool behind the proof is a computationally intense classification of fields in which every multiple of 2 is the sum of squares. We further extend these results to some real cyclotomic fields of large degrees and prove Kitaoka's conjecture for them.
Paper Structure (7 sections, 30 theorems, 54 equations)

This paper contains 7 sections, 30 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Generalities
  3. Multiples of 2 represented as sums of squares
  4. $\mathbb{Z}$-forms and exceptional elements
  5. The quartic case
  6. $2^n$-th roots of unity
  7. $p$-th roots of unity

Key Result

Theorem 1.1

Let $K$ be a totally real number field of degree $d\le 5$. Every element of $2\mathcal{O}_K^+$ is the sum of squares of integers if and only if $K$ is isomorphic to where $\rho$ is a root of $x^3-4x-2.$

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4: O2
  • ...and 44 more