I have a question about algebraic tori, let $T = \mathbb{Q}(i)^\times/\mathbb{Q}^\times$ and let's try to compute $\text{T}(\mathbb{A})/\text{T}(\mathbb{Q})$.
One has a product decomposition for the multiplicative group of Gaussian integers: $$ \mathbb{Q}(i)^\times \simeq \mathbb{Z} \times \prod_{p=4k+1}\mathbb{Z}^2 \times \prod_{p = 4k+3}\mathbb{Z}$$ This has to do with Fermat's theorem that $p = a^2 + b^2$ iff $p = 4k+1$. In that case, with we factor out the multiplicative group of $\mathbb{Q}$ we'd have something like: $$ \mathbb{Q}(i)^\times / \mathbb{Q}^\times \simeq \prod_{p=4k+1} \mathbb{Z}$$ This does not say much about a topolog of $\mathbf{T}$ as a locally compact Abelian group. E.g. this is infinitly generated. We could consider this as a $2 \times 2$ matrix group with elements in $\mathbb{Q}$:
$$ \mathbb{Q}(i)^\times / \mathbb{Q}^\times \simeq \prod_{p=4k+1} \left( \begin{array}{cr} a & -b \\ b & a \end{array} \right)^\mathbb{Z} $$ This is my best guess, and hopefully that's more informative.
Given all this work, Can we give some kind of matrix way of looking at $\mathbf{T}(\mathbb{A})/\mathbf{T}(\mathbb{Q})$ ?