# Expressions unchanged by permuting the roots of a polynomial

I am trying to self-read through Ian Stewart's Galois theory and am stuck at this paragraph. I quote - "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 $\alpha_j$ are permuted. Prominent among these is the expression

$$\delta = \prod_{j< k} (\alpha_j - \alpha_k)$$

which takes just two values, $\pm \delta$, plus for even permutations and minus for the odd ones. Therefore, $\Delta = \delta^{2}$ is a rational function of the coefficients. This gets us started and it yields a complete solution for the quadratic."

My question is as follows: Why should an expression that is unchanged by permuting the roots of a polynomial be expressible as a rational function of the polynomial's coefficients? Is this statement even true or have I misunderstood the book?

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