Splitting fields

From: Joseph (josephsch_at_gmail.com)
Date: 01/28/05 

Date: Fri, 28 Jan 2005 00:52:39 +0000 (UTC)

I came across this little problem, I am not sure if I have answered
it correctly so was looking for a little advise.

It basically asks to prove that Q(2^1/3) is not the splitting field
of any polynomial over Q.

I was thinking that the minimal poly of 2^1/3 is x^3-2, it is also
the smallest monic poly which 2^1/3 as a root, so consquently any
other poly over Q is divisable by the x^3-2.

But x^3-2 has splitting field Q(2^1/3,w) (w=first cubth root of unity) which has degree 6, so from this (not sure if this is right)
we can see that Q(2^1/3) is not the splitting field for any poly

The problem is I am not sure if I have answered the question
correctly or I am missing bits out, or what I have written has
an actual bearing to the question :), could anyone please help

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