Let $\mathbb{Q}(\alpha)$ be a number field, and suppose that $[\mathbb{Q}(\alpha) : \mathbb{Q} ] = m$. Then there are precisely $m$ different embeddings of $\mathbb{Q}(\alpha)$ into $\mathbb{C}$, essentially by sending $\alpha$ to each of its $m$ conjugates.
On the other hand, some of the images of these embeddings can coincide, for instance if $\alpha = i$ (in which case the other conjugate is $-i$, and $\mathbb{Q}(i) = \mathbb{Q}(-i)$).
QUESTION: is there a criterion (for instance in terms of the Galois group of the extension) which tells us precisely how many different images we obtain ?
