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The specific question I'm having trouble with is:

Show X^5-4X+2 has Galois Group S_5 over Q. (I've done this). What is the Galois Group of the polynomial over Q(i)?

I can't do the second part. I'm pretty sure the polynomial is irreducible over Q(i), but that's as much as I can do.

Any help or general tips for dealing with these situations would be appreciated.

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9 
 Dear John, This question will likely be closed soon as being not at a research level. You might try asking it as math.stackexchange.com instead. – Emerton Dec 29 2010 at 22:21
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 A couple of hints: try to convince yourself that the Galois group of your polynomial over any quadratic extension will be either $S_5$ or an index $2$ subgroup of $S_5$. What discriminates between the two cases? – Pete L. Clark Dec 29 2010 at 22:49

closed as too localized by Andres Caicedo, Qiaochu Yuan, Leonid Positselski, Pete L. Clark, Charles Siegel Dec 29 2010 at 22:59

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