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Sums of squares of integer-multiple of an integral element on real bi-quadratic fields

Srijonee Shabnam Chaudhury

Abstract

For any given positive integer $m$ we construct certain totally positive algebraic integers $α$ of a real bi-quadratic field $K$ and obtain some necessary conditions for which $mα$ can not be represented as sum of integral squares. We show this for integers lie in quadratic subfields of $K$ and for integers which are in $K$ but not in any quadratic subfield of $K$. We provide examples in tabular form for each cases to corroborate the results.

Sums of squares of integer-multiple of an integral element on real bi-quadratic fields

Abstract

For any given positive integer we construct certain totally positive algebraic integers of a real bi-quadratic field and obtain some necessary conditions for which can not be represented as sum of integral squares. We show this for integers lie in quadratic subfields of and for integers which are in but not in any quadratic subfield of . We provide examples in tabular form for each cases to corroborate the results.
Paper Structure (11 sections, 22 theorems, 141 equations)

This paper contains 11 sections, 22 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Some Basics
  3. Totally positive integers and totally real field
  4. Bi-quadratic field and its integral basis
  5. Quadratic Forms:
  6. Case I
  7. TABLE I
  8. Case II
  9. TABLE II
  10. Case III and IV
  11. TABLE III

Key Result

Theorem A

(Corollary cor2cor2a) Let $K = \mathbb{Q}(\sqrt{p},\sqrt{q})$ where $p\equiv 2 \pmod 4$ , $q \equiv 3 \pmod 4$. Let $m$, $t \in \mathbb{N}$ and $m \leq$ min $\{ \frac{r}{4 \lceil\kappa \sqrt{r}\rceil}, p, q \}$ for some positive integer $\kappa$ and $F_1 = \mathbb{Q}(\sqrt{p})$, $F_2 = \mathbb{Q}(\s

Theorems & Definitions (34)

  • Theorem A
  • Theorem B
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.1
  • proof
  • Proposition 3.2
  • ...and 24 more