In a question related to introductory Galois theory, I was asked to given an example of a tower of fields F⊂K⊂E such that E is a splitting field for some polynomial f(x)∈F[x], but that K need not be a splitting field. The example that comes to mind for me is Q⊂Q(3√2)⊂Q(ω3,3√2).
So I know that the minimal polynomial for Q(ω3,3√2) over Q is x3−2. This polynomial does not split in the intermediate field Q(3√2).
I wanted to make sure that I'm justifying that this intermediate field is not a splitting field correctly. Now I may be confused about the books terminology. When we say something is not a splitting field, we always have to refer to a specific polynomial correct? I've read a term called a normal extension which I think refers to one in which every polynomial splits, but my book doesn't mention these.
If splitting field means with respect to the polynomial x3−2, then I've already demonstrated that Q(3√2) is not a splitting field. Also, it couldn't be a normal extension either because that polynomial doesn't split.
Does it sound like I'm understanding this correctly or are there some subtleties I might be missing?
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