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Sum of two squares in biquadratic fields

Wenhuan Huang

Abstract

This paper gives an algorithm to determine whether a number in a biquadratic field is a sum of two squares, based on local-global principle of isotropy of quadratic forms.

Sum of two squares in biquadratic fields

Abstract

This paper gives an algorithm to determine whether a number in a biquadratic field is a sum of two squares, based on local-global principle of isotropy of quadratic forms.
Paper Structure (5 sections, 7 theorems, 31 equations)

This paper contains 5 sections, 7 theorems, 31 equations.

Table of Contents

  1. Prime Numbers
  2. General cases
  3. Algorithm about unramified $p$
  4. Algorithm about ramified $p$
  5. Dyadic cases

Key Result

Lemma 1

Let $\mathbb{Q}_2(\sqrt{k})$ be a quadratic extension on $\mathbb{Q}_2$, $k\in\{2,3,7,10,11,14,5\}$, then $t\in \mathbb{Q}_2(\sqrt{k})$ is a square, if and only if ($k=2$)$t=2^mr$, $m\geq0$ an integer, $\sqrt{2}\nmid r$, $r\equiv 1, 3+2\sqrt{2}\pmod{4\sqrt{2}}$. ($k=3$)$t=(\sqrt{3}-1)^{2m}r$, $m\geq

Theorems & Definitions (15)

  • Lemma 1
  • Lemma 2
  • Lemma 1.1
  • proof
  • Remark
  • Lemma 2.1
  • Remark
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 5 more