Explicit image of elements in completion of $\mathbb{Q}(i)$

Question

Let $K = \mathbb{Q}(i)$ and consider a rational prime $p$ such that $p$ splits in $\mathcal{O}_K = \mathbb{Z}[i]$. We then now that if $K_p = K \otimes \mathbb{Q}_p$, we have $K_p \cong \frac{\mathbb{Q}_p[X]}{(X^2+1)} \cong \mathbb{Q}_p \times \mathbb{Q}_p$. My question is if there is any way to know the explicit image of elements in $K_p$ under this isomorphism. For example, if $p = \pi \overline{\pi}$, what would $\pi$ be as an element of $\mathbb{Q}_p \times \mathbb{Q}_p$?

Answer 1

Let $c$ be a square root of $-1$ in $\Bbb{Q}_p$

Then the map $$\Bbb{Q}(i) \to\Bbb{Q}_p\times\Bbb{Q}_p$$ is $$a+ib\mapsto (a+bc, a-bc)$$ Any element of $\Bbb{Q}(i) \otimes_\Bbb{Q} \mathbb{Q}_p$ is of the form $1\otimes a+i\otimes b$ with $a,b\in \Bbb{Q}_p$ and the map $\Bbb{Q}(i) \otimes_\Bbb{Q} \mathbb{Q}_p\to\Bbb{Q}_p\times\Bbb{Q}_p$ stays $1\otimes a+i\otimes b\mapsto (a+bc, a-bc)$.