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).
Bhargava, On the Conway-Schneeberger fifteen theorem, Contemp
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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).