The theorem of symmetric polynomials says:
Observe that for f(x) we have:
The same way we have that:
Since
You want to show that:
Let f (x) = x^4 + px^2 + qx + r ∈ Q[x] be an irreducible quartic. Let α1 , . . . , α4 be the roots of f in a splitting field L over Q. Define elements β1 , . . . , β3 ∈ L by
β1 = (α1 + α2 )(α3 + α4 ),
β2 = (α1 + α3 )(α2 + α4 ),
β3 = (α1 + α4 )(α2 + α3 ).
(a) Show that β1 , β2 , β3 are the roots of the cubic (the so-called resolvent of f )
g(x) = x^3 − 2px^2+(p^2 − 4r)x + q^2 ∈ Q[x].
Any help appreciated. (The hint in the question says to use the theory of symmetric functions...)