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 $\mathbb Q$.)
Torsion elements in $\operatorname{Gal}(\mathbb Q^a /\mathbb Q) $
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abstract-algebra
galois-theory
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0So, that makes only non trivial torsion elements are of order 2, and conjugates of conjugation.. – 2012-11-15
1 Answers
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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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0Dear @Hagen, Regarding the question you mention at the end, this is true: http://mathoverflow.net/questions/138168/embeddings-of-overline-mathbfq-into-mathbfc?lq=1 – 2013-08-15