Abstract

We prove the rank of the group of signatures of the circular units (hence also the full group of units) of $\mathbb{Q}{\left({\zeta }_m\right)}^{+}$ 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 $\mathbb{Q}{\left({\zeta }_{p^n}\right)}^{+}$ 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

Mathematical Subject Classification 2010

Milestones

Authors