This is the summary of my research:
The Ouroboros is referenced in Izumi's statement "All is One, and One is All." The Flamel Snake which is a symbol on Izumi's left breast, at the back of Ed's coat, and Al's houlder is the opposite of the Ouroboros which is the symbol of the homonculi.
This is what I've found on the internet:
In Fullmetal Alchemist, the symbol of the ouroboros is (among other things) what makes the homonculi identifiable, as each of them has an ouroboros tattoed on their bodies. The image of the ouroboros, Greek for "tail swallower", is that of a serpent curled in a circle and devouring its own tail (it is also often depicted with two triangles forming a six-pointed star, as shown here). In a way, the ouroboros is a mockery of the serpent's cross, for both involve snakes and a pair of hovering wings. In the case of the ouroboros, however, these appear more twisted and sinister - the snake is fanged and has two rows of spines along its back, and the wings above it are pointy and bat-like. The Good/Evil theme would be fitting, considering the characters sporting the symbols, though no one in the series is purely "evil", or even purely good.
Like the serpent's cross, the ouroboros is an ancient symbol that is often associated with alchemy. Like the figure eight, it symbolizes infinity or eternity, which is appropriate for the homonculi, who, provided with enough red stones, can theoretically live forever. Aside: In Norse myth, the king Ragnar Lodbrok who had many encounters with snakes, had a son with the image of a white ouroboros around the iris of his eye. This is surprisingly similar to Führer Bradley's "ultimate eye", which bears his tattoo. Perhaps the legend was inspiration?
In alchemy, the ouroboros signified life's inevitable cycle, which had to be overcome for success. It's opposing principles of life and death, sometimes represented by a snake that was half black and half white, were also represented as a balance that must be kept. But if the ouroboros symbolizes the unending cycle of living and dying, so too does it stand for the process of alchemy, which was thought to have no beginning and no end - the finished product was essentially the same as the starting materials.
In an attempt to steer myself away from getting too confusing, I'll move on to the part of the symbolism that I find most interesting because of its close FMA tie-in. The Chrysopoeia of Cleopatra (meaning the "gold making" of Cleopatra - a reference to alchemy) is a volume from the 2nd century c.e. discovered in Alexandria. On one of its pages, there is a drawing of an ouroboros with the words Hen to Pan: "All is One". When I read this, I was very excited, because, as you will know if you have read volume 6 of the manga and/or seen episode 28 of the anime, when Izumi leaves young Ed and Al to fend for themselves on Yock Island, she tells them to find the answer to "All is One, One is All". At the end of the month, the two boys answer their teacher correctly with "All is the world. One is me." This connection is so obvious that it hardly needs explaining. But what else is complete obsession for? The explanation that Hiromu Arakawa gave to the phrase was the idea of continuum and balance in world - both of which are meanings given to the ouroboros. "All is One, One is All" can essentially sum up the ouroboros, for the star within the circle could represent the heavens, and therefore the universe, which is contained within the snake's ring, and yet the snake is also a part of the universe.
