Additively indecomposable quadratic forms over biquadratic and simplest cubic fields

TL;DR

We address the existence and enumeration of additively indecomposable quadratic forms over real biquadratic and simplest cubic fields. The main results show that real biquadratic fields admit classical indecomposable binary forms in $2$ variables, and simplest cubic fields admit indecomposable binary and ternary forms when $ obreak\mathcal{O}_K= obreak\mathbb{Z}[ ho]$, with a lower bound on the number of non-equivalent binary forms. The approach combines indecomposable integers from quadratic subfields, trace and determinant arguments, and explicit constructions, with detailed case analyses including several exceptional subfields. The findings illuminate how indecomposability behavior in subfields extends (or fails to extend) to biquadratic and cubic extensions, and provide quantitative bounds on the abundance of indecomposable binary forms in the simplest cubic setting.

Abstract

In this paper, we study additively indecomposable quadratic forms over real biquadratic and simplest cubic fields. In particular, we show that over these fields, we can always find such a classical form in 2 variables, which differs from the situation for forms over integers. Moreover, for some cases, we derive a lower bound on the number of classical, additively indecomposable binary quadratic forms up to equivalence.

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 3.1: TY
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Theorem 4.1: Man
• Proposition 4.2: TY