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 $\mathcal S^*$ 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 $\mathcal S^*$. 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 $\mathcal S$.