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 November 30th, 2017, 02:57 AM #1 Senior Member   Joined: Jan 2015 From: usa Posts: 101 Thanks: 0 subgroups of a Galois group Please help me to answer the following problem: Let $f(x)=x^6+3\in\mathbb{Q}[x]$. Assume the following facts: 1) $f$ is irreducible in $\mathbb{Q}[x]$. Let $\alpha$ be a root of $f(x)$ and $L=\mathbb{Q}(\alpha)$. 2) $L$ is a splitting field for $f(x)$ over $\mathbb{Q}$. 3) $Gal(L/\mathbb{Q})$ is isomorphic to a copy of $S_3$ inside $S_6$. >For each non-trivial proper subgroup $H  December 19th, 2017, 04:22 AM #2 Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 Have you determined the possible values of$\displaystyle \alpha$. That is, have you determined all 6 roots of$\displaystyle x^6+ 3= 0\$? Last edited by skipjack; December 19th, 2017 at 07:24 PM.

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