In Edwards' "Galois Theory" articles 29-31, the notion of Galois resolvent is motivated by a result of Lagrange (article 104 in his Réflexions sur la résolution algébrique des équations). The theorem reads:
If $t$ and $y$ are polynomials in the roots of a polynomial equation $f(x) = 0$ and if $t$, $y$ are such that every permutation of the roots which changes $y$ also changes $t$, one can, generally speaking, express $y$ rationally in terms of $t$ and the coefficients of $f$.
The "generally speaking" provision restricts the statement to the case where the polynomial $F$ which gathers all the formally distinct polynomial $t, \phi_1t, \dotsc, \phi_kt$: $$F(X) = (X-t)(X-\phi_1t)\ldots(X-\phi_kt)$$ has simple roots, i.e. no two permutations of the roots of $t$ leads to the same numerical value.
The proof is made for this case, but even in Lagrange's text there seems to be a loop between article 100 and 103. Here I just say why I think that Edward's version of the proof seems inconsistent. He says:
Let $t$ and $y$ be given polynomials in the $n$ roots and let $t_1, \dotsc , t_k$ be all the distinct polynomials in the roots that can be obtained from $t$ by permutation of the roots. By assumption, any permutation which leaves $t$ unchanged leaves $y$ unchanged. Therefore, there are at most $k$ different polynomials that can be obtained from $y$ by permuting the variables, and there are polynomials $y_1, \dotsc, y_k$ such that the permutations which carry $t$ to $t_i$ carry $y$ to $y_i$.
The key to Lagrange's theorem is to consider the $k$ polynomials: $$y_1 + \dotsb + y_k ….$$ These polynomials are symmetric and therefore their numerical values are known (expressible in terms of $t$ and $y$ and in terms of the coefficients of the given equation.
However, if two of these $y_i$ are not formally distinct, after permutation they could end up being some $y_j$ and $y_1 + \dotsb + y_k$ would not be symmetric.
