Let $\mathbb{K}$ be a field and let $f \in \mathbb{K}[x]$ be a monic polynomial of degree $n$. Suppose that $\alpha\_1, \ldots, \alpha\_n$ are all the roots of $f$ (in some algebraic closure of $\mathbb{K}$) and define
$$\Delta\_f(y) = \prod\_{\substack{1 \leq i , j \leq n \\ i \neq j}} (y - (\alpha\_i - \alpha\_j)).$$
Clearly, $\Delta\_f(0)$ is the [discriminant](https://en.wikipedia.org/wiki/Discriminant) of $f$, so $\Delta\_f(y)$ can be considered as a generalization of the notion of discriminant of a polynomial. It is also easy to see that $\Delta\_f(y) \in \mathbb{K}[y]$ and that $\Delta\_f(y)$ satisfies the identity 
$$\operatorname{Res}\_x(f(x), f(x + y)) = y^n \Delta\_f(y),$$
where $\operatorname{Res}\_x$ is the resultant respect to $x$.
My questions are:


(1) Does $\Delta\_f(y)$ have a name? Is there any article/book where $\Delta\_f(y)$ is introduced and its properties are studied?


(2) Assume that $\mathbb{K}$ has characteristic 0. If $n = 2$ then $\Delta\_f(y) = y^2 - \Delta\_f(0)$ and it follows that $\Delta\_f(y)$ is irreducible in $\mathbb{K}[y]$ if and only if $f(x)$ is irreducible in $\mathbb{K}[x]$. What can we say about the irreducibility of $\Delta\_f(y)$ for $n \geq 3$?


Thanks for any suggestion/reference.



A generalization of the discriminant of a polynomial
As to question (2): Suppose that $\Delta\_f(y)$ is separable. Then $\Delta\_f(y)$ is irreducible if and only if the Galois group of $f$ acts transitively on the set of differences $\alpha\_i-\alpha\_j$, $i\ne j$, which is equivalent to transitivity on the set of pairs $(\alpha\_i,\alpha\_j)$. This, however, is equivalent to the Galois group of $f$ acting doubly transitively on the roots $\alpha\_i$ of $f$.


