#### 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
##### Abstract

We prove the rank of the group of signatures of the circular units (hence also the full group of units) of $ℚ{\left({\zeta }_{m}\right)}^{+}$ 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 $ℚ{\left({\zeta }_{{p}^{n}}\right)}^{+}$ differs from $\varphi \left({p}^{n}\right)∕2$ 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.

##### Keywords
signature rank of units, cyclotomic fields, abelian extensions, circular units
Primary: 11R18
Secondary: 11R27