What is the symbol on the back of Edward Elric's jacket?

The symbol on the back of his jacket seems to be a symbol common to him, Alphonse, and his teacher.

It is based on Kabbalist and Hermetic alchemy symbolism. It has two major elements, the Winged Crown and the Crucified Serpent

The winged crown symbolizes the mastery or achievement of magisterium over the volatile or some sort of alchemical perfection. The crucified serpent means the stabilization, manifestation or incarnation of the volatile.

Volatility is based on the idea that all things, or material, is made of fire or energy. To stabilize and manifest fire is to give it a form in the physical, or material sphere.

It can be said to be the basis of alchemy in symbolic form, the mastery of fixation of volatile.