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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 $Q^a$? –  Gerry Myerson Nov 15 '12 at 12:33
    
Algebraic Closure of Q. –  Ram Nov 15 '12 at 12:33
    
Using Artin Schreier Theorem, can I say Algebraic Closure of Q is Q(i) hence, Gal group contains only two elements, identity and conjugation? –  Ram Nov 15 '12 at 13:58
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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 $\leq 2$. –  Hagen Nov 15 '12 at 14:06
    
So, that makes only non trivial torsion elements are of order 2, and conjugates of conjugation.. –  Ram Nov 15 '12 at 14:13

1 Answer 1

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? –  Ram Nov 15 '12 at 16:24
    
The argument does not imply that the only automorphism of order $2$ is complex conjugation. –  Hagen Nov 16 '12 at 9:10
    
Dear @Hagen, Regarding the question you mention at the end, this is true: mathoverflow.net/questions/138168/… –  Keenan Kidwell Aug 15 '13 at 20:44

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