TITLE: Discriminant of a monic irreducible integer polynomial vs. discriminant of its splitting field
QUESTION [16 upvotes]: Let $f\in\mathbb{Z}[x]$ be monic and irreducible, let $K=$ splitting field of $f$ over $\mathbb{Q}$. What can we say about the relationship between $disc(f)$ and $\Delta_K$? I seem to remember that one differs from the other by a multiple of a square, but I don't know which is which. On a more philosophical note: why are these quantities related at all? Is there an explanation for why they can be different, i.e. some information that one keeps track of that the other doesn't?

REPLY [13 votes]: I think there was some confusion about the splitting field and the field $\mathbb{Q}[x]/(f(x))$, which is isomorphic to the field generated by one root of $f(x)$. (We always assume that $f(x)$ is monic irreducible.)
Let $\alpha$ be a root of $f(x)$, and $L=\mathbb{Q}(\alpha)$ be the field generated by $\alpha$, $\mathbb{Z}[\alpha]$ be the subring of $\mathcal{O}_L$ generated by $\alpha$. The the discriminant of $f(x)$ is the discriminant of the lattice $\mathbb{Z}[\alpha]$. So $\mathrm{Disc}(f)/\mathrm{Disc}(\mathcal{O}_L)$ is the square of $[\mathcal{O}_L: \mathbb{Z}[\alpha]]$. (See III.3 of Lang's "Algebraic number theory").    
However, for splitting fields, these things hardly compares.  For example, take $f(x)=x^4-x+1$, then the discriminant of $f(x)$ is 229 (a prime, which coincides with the discriminant of the field $L$ in this case) , but the discriminant of the splitting field of $f(x)$ is $229^{12}$ (calculated using Pari), which has 28-digits. (Well, it is not hard to show the discriminant of the splitting field of $f(x)$ share the same prime divisors as the field $L$.)
Sorry about bring up a really old question. It is just I asked myself the same thing today.