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Sum of two squares in cyclic quartic fields

Wenhuan Huang

Abstract

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

Sum of two squares in cyclic quartic fields

Abstract

This paper gives an algorithm to determine whether a number in a cyclic quartic field is a sum of two squares, mainly based on local-global principle of isotropy of quadratic forms.
Paper Structure (2 sections, 18 theorems, 14 equations)

This paper contains 2 sections, 18 theorems, 14 equations.

Table of Contents

  1. Prime numbers
  2. General Cases

Key Result

Theorem 1

$K$ is a (real or imaginary) cyclic quartic extension of $\mathbb{Q}$, if and only if $K=\mathbb{Q}(\sqrt{A(D+B\sqrt{D})})=\mathbb{Q}(\sqrt{A(D-B\sqrt{D})})$, where $A$ is odd and square-free, $D=B^2+C^2$ is square-free, $B>0$, $C>0$, $(A,D)=1$.

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • Lemma 1.3
  • proof
  • Corollary 1.4
  • ...and 15 more