The authors generalize the escalation method to number fields, prove finiteness of criterion sets, and compute, conjecturally and in one case exactly, the analogue of the 15-Theorem over Q(√2), Q(√3), and Q(√5).
Even better sums of squares over quintic and cyclotomic fields
1 Pith paper cite this work. Polarity classification is still indexing.
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abstract
We classify all totally real number fields of degree at most 5 that admit a universal quadratic form with rational integer coefficients; in fact, there are none over the previously unsolved cases of quartic and quintic fields. This fully settles the lifting problem for universal forms in degrees at most 5. The main tool behind the proof is a computationally intensive classification of fields in which every multiple of 2 is the sum of squares. We further extend these results to some real cyclotomic fields of large degrees and prove Kitaoka's conjecture for them.
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Escalations and criteria over real quadratic fields
The authors generalize the escalation method to number fields, prove finiteness of criterion sets, and compute, conjecturally and in one case exactly, the analogue of the 15-Theorem over Q(√2), Q(√3), and Q(√5).