Embeddings of infinite algebraic extensions of $\mathbb Q$

Question

It is known that a totally real number field of degree $n$ has $n$ embeddings to the real numbers.
But what if $K$ is an infinite algebraic extension of $\mathbb Q$?
As my first question, take the field of real algebraic numbers $\mathbb {\bar Q} \cap \mathbb R$. What are, if any, the nontrivial real embeddings?
The second one is the field $\mathbb Q(2^{\frac 1 2}, 2^{\frac 1 4},...)$. I suppose there are no nontrivial real embeddings, because the field $\mathbb Q(2^{\frac 1 4})$ is not totally real, and thus none of its extensions are.
The third one is the complex embeddings of $\mathbb {\bar Q}$. I would think that these are related to the group $Gal(\mathbb {\bar Q}/\mathbb Q)$, but I don't know much about Galois theory.