$\mathbb{Z}$-Bases and $\mathbb{Z}[1/2]$-bases for Washington's cyclotomic units of real cyclotomic fields and totally deployed fields
Rafik Souanef
TL;DR
The paper addresses explicit generation of Washington's cyclotomic units for abelian fields by constructing concrete bases of $\mathbf{Was}(\mathbb{K})/\mathbf{Z}(\mathbb{K})$ and, in the totally deployed case, a $\mathbb{Z}[1/2]$-basis of $\mathbf{Was}_2(\mathbb{K})$. Building on Kučera's cyclotomic bases and Kronecker–Weber conductors, it develops a unified framework to pass from imaginary to real fields, producing detailed basis descriptions for real cyclotomic fields and for totally deployed abelian fields through a careful decomposition by conductor levels and Frobenius/complex-conjugation actions. The main contributions are Theorems 2, 5, and 11, which provide explicit $\mathbb{Z}$-bases in the real case and a $\mathbb{Z}[1/2]$-basis in the totally deployed setting, with Was$_2(\mathbb{K})$ identified as a direct factor of Was$_2(\mathbb{Q}(\zeta_n))$ and connections to Sinnott’s index via Sinnott’s theory and class numbers. These results yield practical generators for unit groups, illuminate the structure of Washington’s units relative to Sinnott’s circular units, and have potential implications for Iwasawa-theoretic investigations and genus-field analyses in abelian extensions.
Abstract
We present families of generators with minimal cardinality - we call such families bases - of the free abelian group $\operatorname{Was}(\mathbb{K}) / \mathbf{Z}(\mathbb{K})$ for any real cyclotomic field $\mathbb{K} = \mathbb{Q}(ζ_n)^{+}$. If $\mathbb{K}$ is a totally deployed abelian number field, we give a $\mathbb{Z} [1/2]$-basis of $\operatorname{Was}(\mathbb{K}) / \mathbf{Z}(\mathbb{K}) \otimes_{\mathbb{Z}} \mathbb{Z} [1/2]$. Here, $\operatorname{Was}(\mathbb{K})$ refers to the group of Washington's cyclotomic units of $\mathbb{K}$ and $\mathbf{Z}(\mathbb{K})$ refers to the group of roots of unity lying in $\mathbb{K}$.