I have read a theorem about the possibility of contructing a segment of given lenght with straightedge and compass. I didn't well understand the theorem, can we use it to deduce the following (equivalent?) result?
Given two orthogonal segments of lengths $a,b$, every segment which that can be constructed form them has a length $c$ such that there is a quadratic polynomial $P(a,b,c)$ with integer coefficient satisfying $$P(a,b,c)=0.$$
or maybe its weaker form,
Given two orthogonal segments of lengths $a,b$, every segment which I can construct form them has a length $c$ such that there is a polynomial $P(a,b,c)$ with integer coefficient satisfying $$P(a,b,c)=0.$$
The first result is false; given two orthogonal segments of nonzero lengths $a$ and $b$, where $a$ is rational and $b$ is irrational, it is possible to construct a segment of length $\sqrt[4]{2}ab$. It is not hard to show that there is no quadratic polynomial $P\in\Bbb{Z}[X,Y,Z]$ such that $$P(a,b,\sqrt[4]{2}ab)=0.$$ The second result is true; every construction step yields a root of a polynomial of degree at most $2$ with coefficients in the field generated by the points constructed so far.