An algebraic number is a solution to a polynomial with rational coeficients over a field $K.$ A transcendental number is a number that is not algebraic.

Has anyone proposed a relative description of algebraic and transcendental numbers?

Working with polynomials $p(x)$ in $\Bbb R^2$ I figured out that I could create infinitely many spaces equivalent to $\Bbb R^2$ under different mappings.

Let's take just one example of a space equivalent to $\Bbb R^2$ and that would be $\Bbb R^2_{\gt 0}.$ The underpinning of the equivalence is the mapping $f:\Bbb R^2 \to \Bbb R^2_{\gt 0}$ where $f(x,y)=(e^x,e^y).$ This mapping allows one to transport information between the spaces, information such as the metric, smooth, field, group, vector space structures etc.

Let $M:=\Bbb R^2_{\gt 0}.$ Since $M$ has only positive ordered pairs we need to recreate what polynomials are in this space. Polynomials in $M$ are related to polynomials in $\Bbb R^2$ by the composition $g(x)=e^{p(\log(x))}$ where $g$ is a polynomial in $M$.

And the general way to convert any function $h(x)$ from $\Bbb R^2$ to $M$ is by exponentiating the expression $h(x)$ and then taking the logarithm of the independent variable, so $j(x)=e^{h(\log(x)}$ where $j$ is the converted function living in $M$.

One consequence of all this is that $g(x)=1$ can attain solutions such as $x=e^2.$ Clearly one expects this number to be a transcendental number, but remember $g(x)$ are our polynomials so in fact $x=e^2$ is an algebraic number in relation to $M$ and furthermore $x=e^2$ is a transcendental number in relation to $\Bbb R^2.$

This is what I mean by a relative description of algebraic and transcendental numbers.

Has anyone proposed a description like this in the literature? Could the formal idea lead anywhere?