I have replaced your [; and ;] with "tex" and "/tex".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?