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In my Galois theory course, some material is developed on normal field extensions and separable field extensions. A couple theorems are proven about them each, and then quickly it is proven that an extension is Galois iff it is normal and separable, and for the rest of the course only Galois extensions are considered.

I can't help but feel like most of that material on separable and normal extensions was just a detour: started off with "useless" definitions for them each (e.g. separable in terms of the way polys split), spend a lot of work proving that they have nicer definitions (e.g. separable in terms of morphisms) and finally proving "Galois = normal + separable" with these nicer definitions.

My question is: what is the advantage in thinking about normal and separable field extensions? I understand we want theorems to have the minimal required assumptions for them to be true, for example the Primitive Element Theorem only needs the assumption that the finite field extension is separable for it to be a simple extension (but this is the only big theorem that comes to mind). Is there really a genuine advantage in thinking about normal and non-separable (or non-normal and separable) extensions?

Tom Misch
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  • See https://math.stackexchange.com/questions/54565/what-is-the-relation-between-normal-extension-and-separable-extension – Wuestenfux Apr 10 '22 at 14:49
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    "Is there really a genuine advantage in thinking about separable (but non-normal) extensions? - Yes, there is. Many theorems of algebraic number theory do not require a Galois extension, but they require "separable". – Dietrich Burde Apr 10 '22 at 15:04
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    Since you asked here about $\mathcal{O}K$, the ring in integers in $K$ - here is one of many theorems, where we need separable, but not normal: *Let $L\supset K$ be a separable extension of global fields and $\alpha\in \mathcal{O}_L$. Then we have $P{\alpha}\in \mathcal{O}K[x]$. In particular, the trace ${\rm tr}{L/K}(\alpha)$ and the norm $N_{L/K}(\alpha)$ are in $\mathcal{O}_K$.* – Dietrich Burde Apr 10 '22 at 15:12
  • Thanks @DietrichBurde, I think I perhaps have such a limited view due to this stuff seemingly coming out of left field in my course and are not really the focus, so maybe I asked a silly question – Tom Misch Apr 10 '22 at 15:17

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