Vol. 298, No. 2, 2019

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Signature ranks of units in cyclotomic extensions of abelian number fields

David S. Dummit, Evan P. Dummit and Hershy Kisilevsky

Vol. 298 (2019), No. 2, 285–298

We prove the rank of the group of signatures of the circular units (hence also the full group of units) of (ζm)+ 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 (ζpn)+ differs from ϕ(pn)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.

signature rank of units, cyclotomic fields, abelian extensions, circular units
Mathematical Subject Classification 2010
Primary: 11R18
Secondary: 11R27
Received: 31 May 2018
Revised: 20 August 2018
Accepted: 21 August 2018
Published: 8 March 2019
David S. Dummit
Department of Mathematics
University of Vermont
Burlington, VT
United States
Evan P. Dummit
School of Mathematical and Statistical Sciences
Arizona State University
Tempe, AZ
United States
Hershy Kisilevsky
Department of Mathematics and Statistics
Concordia University
Montreal, QC