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According to wikipedia, a normal extension is a splitting field of a family of polynomials, and a normal subgroup is one that is invariant under conjugation. Why are normal extensions and normal subgroups "normal"?

user210807
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    The word "normal" for subgroup is not that natural. Earlier there were plenty of other names for it too and in French, e.g., it is "distinguished subgroup" (when translated literally). – quid Jan 26 '15 at 23:22
  • Could you please clarify if you are interested a. in the historical reason for the name b. the relation of the notions c. why they are considered "normal" in some sense. – quid Jan 26 '15 at 23:30
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    However, in characteristic $0$, normal subextensions of a normal extension correspond bijectively to the normal subgroups of its Galois group (it's also true in characteristic $p$, but one has to define Galois extensions). – Bernard Jan 26 '15 at 23:33
  • @quid I'm more interested in the second sense, but the first is interesting too. – user210807 Jan 26 '15 at 23:34
  • @Bernard Yes, the question is partly motivated by the fundamental theorem of Galois theory. – user210807 Jan 26 '15 at 23:35
  • Did the idea of normal subgroups precede normal field extensions or vice versa? – anomaly Jan 27 '15 at 00:02
  • For the etymology of normal subgroups: https://math.stackexchange.com/questions/898977/why-are-normal-subgroups-called-normal – puzzlet Oct 10 '17 at 09:11
  • @anomaly From this answer https://math.stackexchange.com/questions/98456/original-works-of-great-mathematician-%C3%89variste-galois, we can see that when Galois wrote his paper, the concept of field is not there, let along normal field extension. But he did used the idea normal subgroup. So normal subgroup came first. But I didn't have the origin of normal extension yet. – Z Wu Apr 23 '22 at 13:44
  • From what I have found now, Galois has the idea of normal subgroup first. The earliest reference that I found about normal extension is a german reference 1895 textbook [Lehrbuch der Algebra, volumn 2, 89.11]. In that chapter, a relation of normal extension and normal subgroups are mentioned. With the infomation I have, I deduce that normal extension was named after normal subgroups. – Z Wu Apr 23 '22 at 15:28

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