Answer Happy

Topics in Algebra Herstein
Prove this theorem step by step plz
rational THEOREM 5.6.3 Let F be a field and let F(x₁,...,xn) be the field of functions in x₁,...,xn over F. Suppose that S is the field of symmetric rational functions; then 1. [F(x₁,...,x):S] = n!. 2. G(F(x₁,...,x), S) = S, the symmetric group of degree n. 3. If a₁,..., an are the elementary symmetric functions in x₁, xn, then S = F (a₁, A₂,..., an). 4. F(x₁,...,x) is the splitting field over F(a₁,..., an) = S of the polynomial a₁-1 + a₂t"-2...+ (-1)"an th