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.
Galois group of a cubic over the function field $\mathbb Q(y)$
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galois-theory
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2What 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? – 2012-01-29
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0If this is meant to be a polynomial in the variable $x$, then surely $x$ will not appear in its discriminant. Please rethink your question, and figure out what you really mean to ask. – 2012-01-29
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1Aggie is probably considering this as a polynomial in $x$, and made a typo; the discriminant of that polynomial as a polynomial in $x$ is $108(y^3-1)$. – 2012-01-29
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0I am sorry for the typo. David was correct. But I am still lost on how to start this problem. – 2012-01-30
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2@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 $y$? – 2012-01-30
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0If your specific question is «what are all the roots?», as you say, you can use the classical formula for roots of a cubic polynomial, which will give you expressions for the roots in an appropriate extension, no? – 2012-01-30
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0Do you know what the discriminant usually implies about the Galois group of an irreducible cubic? (Whether it's $S_3$ or $A_3$, I mean.) And are we working over $\mathbb Q$? – 2012-01-30
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0Yes we are working over Q. I understand the info about the disc being cubic. But there are 2 variables here. That is what is confusing me. – 2012-01-30
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0@Arturo. I was just given the info above and that the ground field is Q. I have no other info. – 2012-01-30
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1It's more about whether the discriminant is a square. – 2012-01-30
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0Right. My bad, having hungerford memory loss tonight. Now the scalar 108 is a square, but what about $y^{3}$ -$1$? What do I do with that? – 2012-01-30
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1@Aggie: $108$ is not a square. Are you familiar with unique factorization for polynomials? (You don't really need this, but it might be useful.) Can you see why a square polynomial necessarily has even degree? – 2012-01-30
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0I am really jacking this up. Sorry guys. Bad night I guess.Vaguely familar with unique factorization for poly's. – 2012-01-30
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2Out 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