It is well known that ruler and compass cannot trisect an angle. The standard proof of it uses field theory. It is just a contradiction of tower law.
However, I wonder if we can prove this only using geometry axioms for example Hilbert's axioms, introduced in Geometry: Euclid and Beyond by Hartshorne. You can also check the wikipedia page Hilbert's axioms, although these axioms are for 3 dimensions. In this case, ruler and compass construction is still well defined, but the proof using field theory no longer works. Maybe this problem is undecidable? Only with axioms of incidence, order and congruence, the geometry is not the same thing as on a coordinate plane. Are there any ideas or reference on it? Thank you.
To get a sense of what it feels to prove geometry only with axioms, you can also look at my project using lean to formalize Euclidean geometry with Hilbert's axioms, with a page some proofs demonstrating how some of the proofs work. You don't need to know about lean to understand it.
Hilbert's axioms for plane geometry basically capture ruler and compass construction. You can prove that "one cannot show the existence of angle trisection from Hilbert's axioms", using field theory, via the characterization of models, but I don't see how to prove that syntactically, let alone without field theory.
Hartshorne proved in his book that any model $M$ of Hilbert's axioms for plane geometry (including parallel axiom P) is of form $F^2$ where $F$ is a Pythagorean field, and moreover $M$ satisfies CCI (circle-circle intersection) iff $F$ is Euclidean. And then we use field theory to argue that a Pythagorean or even Euclidean field $F$ need not contain enough elements for trisection.