I've been reading Galois' memoir on the insolvability of the quintic and am trying to understand his reasoning on a lemma which seems to be a version of the primitive element theorem.
In modern language, he supposes $K$ is a number field, $a,b, c, ...$ is a finite collection of algebraic numbers, and $V = \varphi(a,b,c,...)$ is a rational function of $a,b,c,...$ for which
$$\varphi(\sigma(a),\sigma(b),\sigma(c),...) \neq \varphi(a,b,c,...) $$
for any permutation $\sigma$ of $a,b,c,...$. Then, Galois claims that $K(V) = K(a,b,c,...)$.
For example, if $V = \sqrt{2} + 2 \sqrt{3}$, then this says that $\sqrt{2}$ and $\sqrt{3}$ both are polynomial functions of $V$.
I'm not able to follow his reasoning (I will supply a picture of the text soon). He says to look at the product
$$F(V,a) = \prod\limits_{\sigma} (V - \varphi(a,\sigma(b),\sigma(c), ...)) = 0.$$
as $\sigma$ runs through all permutations of $b,c,...$. This product is symmetric in $b, c, d, ...$, so Galois says we should just think of this equation as a function of $V$ and $a$. He then says "we can pull $a$ out of the equation." What does this mean, and why does it follow that $a$ (and likewise, $b, c,$ etc.) can be solved for as a rational expression in $V$?