E/E^H is Galois iff all H-orbits on E are finite, and finite Galois iff orbit lengths are bounded; algebraicity of elements holds iff their orbits are finite.
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Galois Extensions via Finiteness of Orbits
E/E^H is Galois iff all H-orbits on E are finite, and finite Galois iff orbit lengths are bounded; algebraicity of elements holds iff their orbits are finite.