Signature ranks of units in cyclotomic extensions of abelian number fields

Abstract

We prove the rank of the group of signatures of the circular units (hence also the full group of units) of Q(ζ m ) + tends to infinity with m. We also show the signature rank of the units differs from its maximum possible value by a bounded amount for all the real subfields of the composite of an abelian field with finitely many odd prime-power cyclotomic towers. In particular, for any prime p the signature rank of the units of Q(ζ p n ) + differs from ϕ(p n )/2 by an amount that is bounded independent of n. Finally, we show conditionally that for general cyclotomic fields the unit signature rank can differ from its maximum possible value by an arbitrarily large amount.

Waste in the US Health Care System

“…For the “failure of care delivery” domain, 11 articles addressed costs of waste (Table 2) and 10 articles addressed potential savings from interventions (Table 3) (2 articles were used for both costs and savings). Cost studies were classified into 3 subcategories: hospital-acquired conditions and adverse events, clinician-related inefficiencies, and lack of adoption of preventive care practices.…”

Higher central charges and Witt groups

“…The signatures of nonzero totally real algebraic numbers, especially those of algebraic units, are studied in number theory [17,16]. It is simple to determine the signatures of real quadratic numbers, but it is a difficult task for higher degree totally real algebraic number.…”

Untitled

“…Originating in the study of solutions to the negative Pell equation, the investigation of signatures of units in number rings dates back at least to Lagrange. While a considerable amount of progress has been made for quadratic fields [7,21,38], predictions for the distribution of narrow class groups and possible signs of units under real embeddings for certain families of higher degree number fields have only recently been developed [4,14,16].…”