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Given the polynomial $x^{3} - 3yx + 2 = 0$ with coefficients in $\mathbb Q(y)$ where $y$ is an indeterminate, which has discriminant $108(y^3 - 1)$, what is the Galois group of this polynomial? Specifically, what are all of the roots? I am having a tough time with this problem. Any suggestions or help would be greatly appreciated.

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    Out of curiosity: how come you know about Galois groups but are only vaguely familiar with unique factorization for polynomials? In most presentations of these subjects, the latter antecedes the former.2012-01-30

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