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October 20, 2025Open Access

Additively indecomposable quadratic forms over biquadratic and simplest cubic fields

Key Points

We identify classical forms in 2 variables that are additively indecomposable over biquadratic fields.
Our evaluation shows that these forms differ significantly from those over integers, revealing unique properties.
We derive a lower bound on the equivalence classes of binary quadratic forms for specific cases.
This research advances understanding of quadratic forms particularly in the context of cubic fields.

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.

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Cite This Study

Fryšová et al. (Sun,) studied this question.

synapsesocial.com/papers/68f64fbb2509bc8625bfb206 https://doi.org/https://doi.org/10.48550/arxiv.2509.23857

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