Consider a motivating example:
Let $E\in \mathbb{Q}[y][x]$ be of degree $n=2$ (in $x$) and separable when viewed as a member of $\mathbb{Q}[x,y]$. Therefore we can calculate it's roots in $\mathbb{Q}[y][\sqrt[n]{D_E(y)}]$[1] where $D_E(y)\in\mathbb{Q}[y]$ is the discriminant of $E$
If we look for rational solutions $(x,y)$ for $E$ we must have $D_E(y)=z^n$ for some rational $z$ and therefore we can see that every such solution gives rise to a rational point $(y,z)$ on the curve $E': z^n=D_E(y)$.
Now this is not a bijection on the rational points of the curves, so we might as well make it one:
We can express $E'$ as a function of $y$ and $x(y,z)$ only (due to the fact that $D_E$ is a symmetric function of the roots) and writing $w=x(y,z)$ we can get a new curve $E''(y,w)$ with the same rational points as $E$.
[1] We can do all of the above in degrees $3, 4$ as well, only we would have to repeat the construction a number of times. We only need the solvaibilty of $S_n$ for $n\leq 4$ for the construction.
Example:
Let $E$ denote $x^2+3xy^2+2y^4+4$
$\Rightarrow D(y)=y^4-16$
$E(F)=\{(\frac{-3y^2+\sqrt{D(y)}}{2}, y) \cup (\frac{-3y^2-\sqrt{D(y)}}{2}, y): y\in F\}$ for every field $F/\mathbb{Q}(y)(\sqrt{D(y)})$
Now $(x,y)\in \mathbb{Q}^2 \iff (y, \sqrt{D(y)})\in\mathbb{Q}^2$
And we can now write $z^2 = y^4 - 16$ and we want different $z$s that give rise to the same $x$ to yield the same solution. We can see that if $x(y,z) = x(y, z')$ then we must have either $z=z'$ or $z'=-z$ and therefore we care about solutions to $z^2-y^4+16$ up to the sign of $z$ which we know how to solve (this is a pet example as we could solve both equations without doing this either way).
Now I want this this to lift this operation to an endomorphism on some subspace of curves with $E \mapsto E''$ (that preserves all rational points), the problem is that we seem to not have much control on the degree of $E''$ (in fact it seems to increase so this surely can't work in the present form).
All the above construction is just repeated application of basic Galois theory on the ground field $\mathbb{Q}(x_1,...,x_n)$
Question: Can this be replicated in some other ground field to make the following a well defined operation on some subspace of curves (and still preserve points in some "field of definition", or at least say these points lie over each other, if this field of definition is not the same between $E$ and $E''$)?
This comes from playing around with some examples, and I might be missing something basic that would not make this possible, so the answer to the question is probably "only trivial classes", but maybe there is interesting theory behind this.
I think this can only apply in some generalization of "theory of solvability by radicals" but am not sure what nature should this have.
