A classification of $\mathbb{Q}$-valued linear functionals on $\overline{\mathbb{Q}}^{\times }$ modulo units
Volume 205 / 2022
Abstract
Let $\overline{\mathbb Q}$ be an algebraic closure of $\mathbb Q$ and let $\overline{\mathbb Z}$ denote the ring of algebraic integers in $\overline{\mathbb Q}$. If $\mathcal V = \overline{\mathbb Q}^\times /\overline{\mathbb Z}^\times $ then $\mathcal V$ is a vector space over $\mathbb Q$. We provide a complete classification all elements in the algebraic dual $\mathcal V^*$ of $\mathcal V$ in terms of another $\mathbb Q$-vector space called the space of consistent maps. With an appropriate norm on $\mathcal V$, we further classify the continuous elements of $\mathcal V^*$. As applications of our results, we classify extensions of the prime Omega function to $\mathcal V$ and discuss a natural action of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)$ on $\mathcal V$. All results are closely connected to a 2009 article of Allcock and Vaaler on the vector space $\mathcal G = \overline{\mathbb Q}^\times /\overline{\mathbb Q}^\times _{\mathrm{tors}}$.
