Thank you for sharing your notes—everyone needs a different push/viewpoint to make key concepts click, and, the more expositions are out there, the more likely someone will be to find the one that's right for them!
Is M^* Stewart's notation? I'm much more used to seeing it used for a dual space than a group of automorphisms. Conventions aside, it's a bit unfortunate in that it doesn't specify the field on which we're acting! I think that the notation Aut(L/M), or Gal(L/M) for a Galois extension, is more common.
Thank you! Yes, M^* is Stewart's notation. Indeed this notation is "lossy" in the sense that it loses information about what the extension field is. The extension field is usually introduced once for every example or theorem and then M^* is used as a shorthand notation. For example, in my post, the extension field is introduced like this:
"The notation M^* denotes the group of all M-automorphisms of L with composition as the group operation."
By the way, Stewart uses the notation Γ(L/M) too at several places but in this specific lemma, the notation M^* is used and therefore my post too uses the same notation.
Is M^* Stewart's notation? I'm much more used to seeing it used for a dual space than a group of automorphisms. Conventions aside, it's a bit unfortunate in that it doesn't specify the field on which we're acting! I think that the notation Aut(L/M), or Gal(L/M) for a Galois extension, is more common.