Additively indecomposable quadratic forms over biquadratic and simplest cubic fields

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.

💡 Research Summary

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The paper investigates the existence and abundance of additively indecomposable quadratic forms over two families of totally real number fields: real biquadratic fields and the so‑called simplest cubic fields introduced by Shanks. An element α of the ring of integers O_K is called indecomposable if it cannot be written as a sum of two totally positive algebraic integers. Extending this notion, a quadratic form Q is additively indecomposable when it cannot be expressed as a sum Q = Q₁ + Q₂ of two non‑zero totally positive semidefinite quadratic forms (all forms are assumed classical, i.e., off‑diagonal coefficients are even).

The authors first recall basic notation (trace, norm, total positivity, ordering on O_K) and define classical quadratic forms, their Gram matrices, and determinants. They then present a key structural result (Proposition 3.1, originally from