The field extension lattice is isomorphic to the subgroup lattice; if you really wanted to, you could write this out symbolically (but I am not sure why you would want to, since it does not really convey the meaning of the theorem any better). I suppose you might say that such an isomorphism does not qualify as an equation, but that is a bit pedantic in my opinion since such isomorphisms have all the properties of an equivalence relation.

Right, it's a beautiful theorem, not so much a beautiful equation.

Euler's identity is a beautiful equation, because it ties together several of the most fundamental objects of mathematics, with one occurrence of each, with no wasted boilerplate. The notation is part of the beauty. It looks darn good, on the surface in addition to the beyond the ideas behind the surface.

Good question. Though I suppose it's not too difficult to state in terms of the Abel-Ruffini theorem, but then again that watered down version would probably fail to mention the wide-reaching consequences of Galois...