Motivation
If you were asked to write down the coordinates of a set of four points in $\mathbb{R}^3$ that form a regular tetrahedron, you might come up with
$$ \left\{(0, 0, 0),(1,0,0),\left(\frac{1}{2},\frac{\sqrt{3}}{2},0\right),\left(\frac{1}{2},\frac{\sqrt{3}}{6},\frac{\sqrt{6}}{3}\right)\right\}. $$
This is pretty good: all of the coordinates are algebraic, and the largest degree is 2. But you can represent a regular tetrahedron with a simpler set of coordinates:
$$ \{(1,1,1),(1,-1,-1),(-1,1,-1),(-1,-1,1)\} $$
Every coordinate is rational! Wouldn't it be neat if you could input the first set of coordinates into some algorithm, and the algorithm would transform it into the second set, or at least some set of coordinates where every coordinate was rational? Of course, for many input sets there wouldn't be a rational representation, so let's aim for minimizing the largest degree that appears.
Unfortunately, this seems pretty hard (to me) even in one dimension, so that's what I'll focus on for my question.
The question
Let $S$ be a set of real algebraic numbers, for example
$$ \{(8x^3+4x^2-10x-37)_0, (x^6+4x^5-26x^4-90x^3+200x^2+100x-775)_1, (x^6-2x^5-22x^4+22x^3+142x^2-272x-131)_1\}. $$
Let the degree of $S$ be the largest degree of any element of $S$. In my example, $\deg S = 6$.
Let $T$ be a function $x\mapsto ax + b$, with $a$ and $b$ real algebraic and $a$ nonzero. How can I choose $a$ and $b$ to minimize $\deg(T(S))$? In my example, I can choose $a=1$ and $b=(x^3-2x^2+5)_0$ to get
$$ T(S) = \left\{\frac{1}{2},\sqrt{10},1+\sqrt{7}\right\}, $$
and $\deg(T(S)) = 2$. This is the lowest degree for any $T$. As you can probably guess, I constructed this example backwards, starting with $T(S)$. I have no idea how to go in the forward direction.
In the higher-dimensional generalization, $T$ is an invertible similarity transformation on $(\mathbb{R}\cap\overline{\mathbb{Q}})^n$.
Notation
Applying $T$ to a set means applying it to every element of the set.
$(p(x))_k$ means the $k$th real root of $p(x)$, in increasing order, starting at $k=0$.