I've posted this [to Math.SE](https://math.stackexchange.com/q/2733614/214353) about a month ago:


Seems like
$$
\Delta(a\_0+a\_1t^d+a\_2t^{2d}+...+a\_nt^{nd})=(-1)^{n\frac{d(d-1)}2}d^{nd}(a\_0a\_n)^{d-1}[\Delta(a\_0+a\_1t+a\_2t^2+...+a\_nt^n)]^d,
$$
where $\Delta$ is the discriminant.


Presumably this is not difficult to prove, but I just need a reference. Who did discover this formula for the first time?



What would be a standard reference for the formula of the discriminant of $f(t^d)$?
You say "presumably not difficult to prove." Did you try? It seems like a pretty easy exercise, using
$$ \Delta(F(t)) = \prod\_{F(a)=0} F'(a). $$
Taking $F(t)=f(t^d)$, the roots of $F$ are the $d$'th roots of the roots of $f$, while $F'(t)=dt^{d-1}f'(t^d)$. So I doubt you'll find this formula in a reference, but if you need to use it in a paper, you can just state it as a lemma, and for the proof, say "Exercise using the formula $ \Delta(F(t)) = \prod\_{F(a)=0} F'(a) $ with $F(t)=f(t^d)$."


