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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.

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    $\begingroup$ Hint: In the case of $K=\mathbb Q(2^{\frac 1 2}, 2^{\frac 1 4},...)$ you can use the fact that all the numbers $2^{1/2^k}$ are squares of other elements of $K$. Therefore in real embedding they must be mapped to positive numbers (do you see why?) implying that... $\endgroup$ – Jyrki Lahtonen Mar 4 '18 at 20:27
  • $\begingroup$ @JyrkiLahtonen Thanks! I can see now. $\endgroup$ – FusRoDah Mar 4 '18 at 20:58

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