# projectors in a tensor product of number fields

Let $F=\mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic number field and choose an ordering of the Galois group $Gal(F/\mathbb{Q})$, let us say $\{id, \sigma\}$. Then one has an isomorphism

$F \otimes_\mathbb{Q} F \stackrel{\sim}{\longrightarrow} F \oplus F$

sending the elementary tensors $\alpha \otimes \beta$ to $(\alpha\beta, \alpha\sigma(\beta))$.

Can anybody explain me how to construct the preimages of the projectors $(1, 0)$ and $(0, 1)$?

Hint: Look at the images of $(\sqrt{-d}\otimes \sqrt{-d}) \pm (1 \otimes d)$.