Compatibility of the existence of ether and Galilean relativity Why does the idea of ether contradict Galilean relativity? Consider for example what Coiller wrote in his book, A Most Incomprehensible Thing - Notes Towards a Very Gentle Introduction to the Mathematics of Relativity:

...if the light only traveled with its ‘correct’ speed c with respect to the ether frame, then this frame would constitute an absolute frame of reference. An absolute frame is a preferred frame. An inertial observer would then, by using the Galilean transformations, be able to calculate (assuming that they could ever get the experiments to work!) the relative velocity of their own frame of reference. Inertial frames could then be distinguished from each other, and bye-bye Galilean relativity.

How is ether frame absolute? Sure it covers whole space but it's still something within space and thus not what Newton meant when he used the word absolute, so it's not any more different than being able to tell if the sound-producing observer is the one doing the movement or you in an air field.
Also if one still prefers to call ether frame an absolute frame, then why don't we treat the "fixed stars" as a measure of the absolute character of space too in inertial situations, because here too we can tell if we are moving with respect to the rest of the universe or not.
 A: You are right that, in principle, there is no contradiction. In fact, ether was invented to preserve Galilean relativity. Galilean relativity says that the same isolated experiment, performed in different inertial frames, yields the same results.
But if there exists an ether frame, our experiment will no longer be the same in a different frame. It'll be a different experiment even if we arrange the particles the same way initially. This is because the ether is part of the experiment and it'll have different speeds in different frames. So the experiments will not have the same result because the experiments will be different.
But as long as everything transforms the Galilean way, there is no mathematical contradiction. There is only a philosophical contradiction that the rest frame of ether will be considered "preferable" because Maxwell's equations take the simplest form there.
In fact, back then it was considered that Maxwell's equations were incomplete. If the complete version made the ether fluid move-able, then there would be no philosophical contradiction either.
P. S. I'm not sure about the last paragraph because movements in the ether fluid are light itself. The idea I meant is that, suppose a uniform speed car is carrying a fish tank. Then waves in the fish tank water would behave the exact same way as waves in a water tank at rest in Earth's frame.
If we consider parts of the ether fluid as move-able in tank-like blocks, then for someone at rest wrt some ether tank, the waves there will behave the same way as given by Maxwell's equations.
