Abstract
We study the structure of additively indecomposable integers in families of totally real cubic fields. We prove that for cubic orders in these fields, the minimal traces of indecomposable integers multiplied by totally positive elements of the codifferent can be arbitrarily large. This is very surprising, as in the so-far studied examples of quadratic and simplest cubic fields, this minimum is 1 or 2. We further give sharp upper bounds on the norms of indecomposable integers in our families.
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Acknowledgements
The author wishes to express her thanks to Vítězslav Kala for many helpful comments. The author also gratefully acknowledges the many helpful suggestions of anonymous referee.
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The author was supported by Czech Science Foundation GAČR, grants 21-00420M and 22-11563O, by projects PRIMUS/20/SCI/002, UNCE/SCI/022, GA UK 1298218 from Charles University, and by SVV-2020-260589.
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Tinková, M. Trace and norm of indecomposable integers in cubic orders. Ramanujan J 61, 1121–1144 (2023). https://doi.org/10.1007/s11139-022-00669-y
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DOI: https://doi.org/10.1007/s11139-022-00669-y