A classification of $\mathbb{Q}$-valued linear functionals on ${\overline{\mathbb{Q}}}^{\times }$ modulo units

Abstract

Let $\overline{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$ and let $A$ denote the ring of algebraic integers in $\overline{\mathbb{Q}}$. If $\mathcal{S}={\overline{\mathbb{Q}}}^{\times }/A^{\times }$ then $\mathcal{S}$ is a vector space over $\mathbb{Q}$. We provide a complete classification all elements in the algebraic dual ${\mathcal{S}}^{\ast }$ of $\mathcal{S}$ in terms of another $\mathbb{Q}$-vector space called the space of consistent maps. With an appropriate norm on $\mathcal{S}$, we further classify the continuous elements of ${\mathcal{S}}^{\ast }$. As applications of our results, we classify extensions of the prime Omega function to $\mathcal{S}$ and discuss a natural action of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on $\mathcal{S}$.

21 pages

References in corpus (1)

• Direct Limits of Adèle Rings and Their Completions