I have a bunch of assertions without motivation which I'm trying to sort out.

Let H be a subgroup of S_n,

, and

the elementary symmetric polynomials. The assertions are:

i) A_H is defined as the integral closure of

in

, and

.

ii)

is the field of fractions of A_H, "i.e."

.

Alright, so about the first assertion, that A_H is the intersection. This intersection contains only polynomials, i.e. we must have

, so we can simply consider the intersection

, which must be

. I've no idea how to show or see that

is the supposed integral closure, but let's leave it at that for the moment.

Second assertion. What the notation

means I have no idea; my guess is that it's the set

. If this is the case, how do I verify it? Because I would have thought that Frac A_H = k(x_1,...,x_n)^H?