Galois theory
Euclid had wanted to find a way of cutting any angle in thirds. Galois showed that if a ruler and compass could be used to make a third of an angle, it would have an associated equation of degree 3. He showed that only powers of two could be made, by analyzing the equation's group and field. Galois showed that trisecting an arbitrary angle was impossible. Galois' theory could also be used to show that it was impossible to square a circle and double a cube.