# Galois group of a cubic over the function field $\mathbb Q(y)$

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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What does "Galois group of a polynomial" mean when the polynomial is a polynomial in more than one variable? What does "splitting field" of a multi-variable polynomial mean? – Arturo Magidin Jan 29 '12 at 21:53
Aggie is probably considering this as a polynomial in $x$, and made a typo; the discriminant of that polynomial as a polynomial in $x$ is $108(y^3-1)$. – David Speyer Jan 29 '12 at 23:54
@Aggie: What is the ground field? Are you considering this as a polynomial over $\mathbb{Q}(y)$ and asking for its Galois group over that group? Are you considering this as a family of polynomials over $\mathbb{Q}$ parametrized by $y$ (ranging where?) and asking for a description of the Galois group parametrized by $y$? – Arturo Magidin Jan 30 '12 at 0:33
It's more about whether the discriminant is a square. – Dylan Moreland Jan 30 '12 at 0:52
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. – Mariano Suárez-Alvarez Jan 30 '12 at 2:59