What do these adeles look like over $\mathbb{Q}(i)$? These are simply fractions where the numerator and denominator are allowed to have the number $i = \sqrt{-1}$. The elements look like:
$$ \left\{ \frac{a + bi}{c+di}: a,b,c,d \in \mathbb{Z}, \text{ but } c+di \neq 0 \right\}$$
For now I am sweeping under the rug all the details about how we can enumerate all these fractions, e.g. by the Calkin-Wilf tree.
Today's question is about the Adeles over $\mathbb{Q}(i)$ and how to write them down if we stuck with the rationals:
$$ \mathbb{A} = \{(x_\infty, x_2, x_3, x_5, \dots, ) : \text{"most" } |x|_p = 1\} \subset \mathbb{R}\times \mathbb{Q}_2 \times \mathbb{Q}_3 \times \mathbb{Q}_3 \times \dots $$
Here most here means all but finitely many, so the adeles $\mathbb{A}$ are a restricted product over all $p$-adic numbers, $\mathbb{A} = \prod' \mathbb{Q}_p$.
If we work over the extension field $\mathbb{Q}(i)$ there are other primes. Certain primes split (such as $5 = (2+i)(2-i)$) and others ramify (such as $2 = (1+i)(1-i)$). Are the adeles "just" the product over all the Gaussian primes then?
$$ \mathbb{A}_{\mathbb{Q}(i)} = \prod'_{p = 4k+3} \mathbb{Q}_p \times \prod'_{a^2 + b^2 = p} \mathbb{Q}_{a+bi}$$
Do we have to include both $a+bi$ and $a - bi$?