Galois theory and Lagrange's method on quintics

Question

I have just begun studying Galois Theory from Stewart's book and got some questions with some conclusions regarding Lagrange's methods when discussing the quintics. For reference, I will put below part of the discussion in the book:

Lagrange observed that all methods for solving polynomial equations by radicals involve constructing rational functions of the roots that take a small number of values when the roots αj are permuted. Prominent among these is the expression

δ = ∏_1≤j<k≤n (α_j − α_k)(1.13)

where n is the degree. This takes just two values, ±δ: plus for even permutations and minus for odd ones. Therefore ∆ = δ² (known as the discriminant because it is nonzero precisely when the roots are distinct, so it ‘discriminates’ among the roots) is a rational function of the coefficients. This gets us started, and it yields a complete solution for the quadratic, but for cubics upwards it does not help much unless we can find other expressions in the roots with similar properties under permutation.

Lagrange worked out what these expressions look like for the cubic and the quartic, and noticed a pattern. For example, if a cubic polynomial has roots α₁, α₂, α₃, and ω is a primitive cube root of unity, then the expression

u = (α₁ + ωα₂ + ω²α₃)³

takes exactly two distinct values. In fact, even permutations leave it unchanged, while odd permutations transform it to

v = (α₁ + ω²α₂ + ωα₃)³

It follows that u+v and uv are fixed by all permutations of the roots and must therefore be expressible as rational functions of the coefficients. So u + v = a, uv = b where a, b are rational functions of the coefficients. Therefore u and v are the solutions of the quadratic equation t² − at + b = 0, so they can be expressed using square roots. But now the further use of cube roots expresses α₁ + ωα₂ + ω²α₃ = ³√u and α₁ + ω²α₂ + ωα₃ = ³√v by radicals. Since we also know that α₁ + α₂ + α₃ is minus the coefficient of t², we have three independent linear equations in the roots, which are easily solved.

I have a couple of questions:

1 - ∆ can be expressed as a rational function as a result of the fundamental theorem on symmetric functions?

2 - Why can u+v and uv be expressed as rational functions?

3 - How the author got the conclusion that α₁ + α₂ + α₃ = -1?

Answer 1

1. Yes, $\Delta$ is a polynomial in $\alpha_i$ that is invariant under all permutations of them ($\delta$ isn't).
2. An even permutation sends $u$ to $u$ and $v$ to $v$, and an odd one interchanges them.
3. Vieta's formulas (just expand the product $\prod\limits_{i = 1}^3 (t - \alpha_i)$).