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arXiv · 2107.11772

A cubic ring of integers with the smallest Pythagoras number

Jakub Krásenský
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Abstract

We prove that the ring of integers in the totally real cubic subfield $K^{(49)}$ of the cyclotomic field $\mathbb{Q}(\zeta_7)$ has Pythagoras number equal to $4$. This is the smallest possible value for a totally real number field of odd degree. Moreover, we determine which numbers are sums of integral squares in this field, and use this knowledge to construct a diagonal universal quadratic form in five variables.

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Jakub Krásenský. 2021-07-25. A cubic ring of integers with the smallest Pythagoras number. https://doi.org/10.1007/s00013-021-01662-5

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