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arXiv · 2212.00364

Additive structure of non-monogenic simplest cubic fields

Daniel Gil-Muñoz·Magdaléna Tinková
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Abstract

We consider Shanks' simplest cubic fields $K$ for which the index $[\mathcal{O}_K:\mathbb{Z}[\rho]]$ of a root $\rho$ of the defining parametric polynomial is $3$. For them, we study the additive indecomposables of $K$ and provide a complete list of them. Moreover, we use the knowledge of the indecomposables to prove some interesting consequences on the arithmetic of $K$. Mainly, we obtain good bounds on the ranks of universal quadratic forms over $K$ and prove that the Pythagoras number of $\mathcal{O}_K$ is $6$.

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Daniel Gil-Muñoz, Magdaléna Tinková. 2022-12-01. Additive structure of non-monogenic simplest cubic fields. https://doi.org/10.1007/s11139-024-01005-2

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