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Let $k$ be a field and $f(x)\in k[x]$. Let $g(x) = f(\alpha x + \beta)$ for some $\alpha, \beta \in k, \alpha\neq 0$. Prove that $f(x)$ and $g(x)$ have the same discriminants and Galois groups.

I have evaluated the case for when the discriminant is 0, but I'm confused as where to go with the non-zero case.....

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    I mentioned it to my prof and the statement that the discriminants are the same is false. Thank you everyone for the input2012-11-20

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Up to a sign the discriminant is the product of the differences of the roots.

If $r_i$ are the roots of $f$ the the roots of $g$ are $(r_i-b)/a$ so the differences are $(r_i-r_j)/a$.

Thus the discriminant of $g$ is $1/a^m$ times the discriminant of $f$, where $m=n(n-1)$, and $n$ is the degree.

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    @MikeM. you're right2012-11-19