The equation that couldn't be solved
Galois work was the start of Group theory, which is the mathematical description of symmetry. Livio goes on to show the important part this theory has come to play in the study of physics, for instance Noether's theorem which relates symmetries to conservation laws, and the use of group theory in particle physics.
My one criticism of the book is that sometimes Livio seems to discuss a subject just of the sake of it. Of course symmetry gives a wide scope for discussion, and some people might enjoy the variety, but I felt that a more focussed approach would have been better.