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A cubic ring of integers with the smallest Pythagoras number

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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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  1. Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18600, Praha 8, Czech Republic

    Jakub Krásenský

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  1. Jakub Krásenský
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Correspondence to Jakub Krásenský.

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The author aknowledges partial support by project PRIMUS/20/SCI/002 from Charles University, by Czech Science Foundation GAČR, grant 21-00420M, by projects GA UK No. 742120 and UNCE/SCI/022 from Charles University, and by SVV-2020-260589.

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Krásenský, J. A cubic ring of integers with the smallest Pythagoras number. Arch. Math. 118, 39–48 (2022). https://doi.org/10.1007/s00013-021-01662-5

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  • DOI: https://doi.org/10.1007/s00013-021-01662-5

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