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November 30th, 2017, 03:57 AM  #1 
Senior Member Joined: Jan 2015 From: usa Posts: 103 Thanks: 1  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 nontrivial proper subgroup $H<Gal(L/\mathbb{Q})$ determine $Fix(H)$, Express each answer in the form $\mathbb{Q}(\beta)$ with $\beta$ given explicitly in terms of $\alpha$ and also give the minimal polynomial of $\beta$ over $\mathbb{Q}$ Last edited by skipjack; December 19th, 2017 at 08:25 PM. 
December 19th, 2017, 05:22 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 08:24 PM. 

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