TITLE: How do I calculate the discriminant of a galois closure and its other subfields?
QUESTION [8 upvotes]: Given a number field K of dimension d over Q, and galois closure of dimension d! over Q (i.e galois group Sd), can we relate the discriminant of the galois closure to that of the discriminant of K? Assume no special ramification happens in the closure or the other subfields, for example if the discriminant of K is a prime.
Tests in sage indicate that the discriminant of the galois closure is $\Delta_K^{\frac{d!}{2}}$, and that the discriminants of the other subfields are also powers of $\Delta_K$, but the power has something to do with the corresponding subgroup of Sd. (Not just the size of the subgroup)
Is there a way to prove the first indication, and thoughts about the second?

REPLY [3 votes]: The problem of computing the relative discriminant of an extension 
of number fields can be reduced to the computation of the relative
discriminant of a local kummerian extension of degree equal to the residual
characteristic, which is easy to solve.  See for example arXiv:0711.3878v1 [math.NT], p. 51.