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How to find non trivial Torsion elements in $Gal( \mathbb Q^a /\mathbb Q) $, one element will be Conjugation, is there exist any other non trivial Torsion elements in the group $Gal( \mathbb Q^a /\mathbb Q) $. $\mathbb Q^a$ is the algebraic closure of $\mathbb Q$ |
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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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