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How to find non trivial torsion elements in $\operatorname{Gal}(\mathbb Q^a /\mathbb Q) $? One element will be conjugation, but is there any other non trivial torsion element? (Here $\mathbb Q^a$ denotes the algebraic closure of $\mathbb Q$.)

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    I assume $Q$ is the rationals, but what is $Q^a$?2012-11-15
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    Algebraic Closure of Q.2012-11-15
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    Using Artin Schreier Theorem, can I say Algebraic Closure of Q is Q(i) hence, Gal group contains only two elements, identity and conjugation?2012-11-15
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    No. Q(i) is not algebraically closed. What you can say based on the Artin-Schreier-theorem is that any torsion element of $Gal( \mathbb Q^a /\mathbb Q)$ has an order $\leq 2$.2012-11-15
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    So, that makes only non trivial torsion elements are of order 2, and conjugates of conjugation..2012-11-15

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Let $\sigma$ be a torsion element of $Gal( \mathbb Q^a /\mathbb Q)$, that is it generates a finite subgroup $G$. Let $F$ be the fixed field of this group. Then $\mathbb{Q}^a/F$ is finite, hence is of degree $2$ by the Artin-Schreier-theorem. So $\sigma$ has order $2$.

Note that every conjugate of an element of order $2$ has order $2$. So the question arises whether two elements of order $2$ are conjugate, and thus whether the torsion elements are precisely the conjugates of complex conjugation.

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    does this imply only one torsion element exits or I am wrong?2012-11-15
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    The argument does not imply that the only automorphism of order $2$ is complex conjugation.2012-11-16
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    Dear @Hagen, Regarding the question you mention at the end, this is true: http://mathoverflow.net/questions/138168/embeddings-of-overline-mathbfq-into-mathbfc?lq=12013-08-15