In FMA, the back of Edward's coat shows this symbol:
I can't figure out what it would be a reference to in-universe, because the symbology seems Judeo-Christian (a cross and what looks like it could be a serpent), and the show doesn't really have any Christian symbology in it. Is this some symbol in real life? What is it supposed to mean?
It's called the Flamel (yes, named after Nicolas Flamel, purported to be an alchemist who sought the Philosopher's Stone).
It is most noticeably similar to the symbol of Hermes, a god of alchemy, and also the Rod of Asclepius, which you may recognize from ambulances as the EMS symbol.
It is not a religious symbol, nor is it ever explained to have a relevance in the FMA universe. However, given its real life links to alchemy and medicine, it's safe to say that the author intended to bring forth real-world alchemical symbols into her universe.
Further reading: FMA Wiki entry on the Flamel
Its a Flamel
The Flamel refers to the mystical and alchemical symbol depicted as a cross with a snake or serpent draped about it, as well as detached wings and a crown above it.
In Alchemy, the Flamel represents the "fixing of the volatile", a vital step in the alchemical opus' process, related to the making of the mercury's elixir and of curative processes.
The Flamel is named after Nicolas Flamel Whom After his death developed a reputation as an alchemist. However, these legendary accounts only appeared in the seventeenth century.
"Flamel was a real person, and he may have dabbled in alchemy, but his reputation as an author and immortal adept must be accepted as an invention of the seventeenth century." - Dixon, Laurinda
Well as everybody on here has said, it's a flamel which is a symbol created by the alchemist Nicholas Flamel.
After looking into it a bit more, I learned that Flamel often used this symbol when he talked about the Elixir of Life in his books.
So that makes it a fitting symbol for Dante, as she thought she had discovered immortality.
As far as the wings and crown, I'm not sure what those represent, as those are not part of the original Flamel symbol. I'm guessing its something Dante added to it, to make it her own. But like I said, I'm not sure what it represents.
I found a bible verse similar to this in Numbers 21:8, stating
And the LORD said to Moses, “Make a fiery serpent and set it on a pole, and everyone who is bitten, when he sees it, shall live.”
Another interpretation is "to be pure, to be purified". I say this because in FMA, alchemy is what the world revolves around. Even in our reality, alchemy was fabled to transmute any metal into gold. Since gold is considered to be the purest of all metals, we can also transcribe that context to the purest (or purified) form of expression, ideas, self, etc.
Though I know where you were going with the Christian symbolism. The snake on the rod (also a root that the medical symbols came from because People were healed by looking at the snake on the rod)
It is also said to be the crucifixion of the serpent(AKA the natural man) which sort of happens to Ed as the series progresses.
