Tuesday, 1 September 2015

abstract algebra - Help justifying that mathbbQ(sqrt[3]2) is not a splitting field over mathbbQ.

In a question related to introductory Galois theory, I was asked to given an example of a tower of fields FKE 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 QQ(32)Q(ω3,32).



So I know that the minimal polynomial for Q(ω3,32) over Q is x32. This polynomial does not split in the intermediate field Q(32).



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 x32, then I've already demonstrated that Q(32) 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?

No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...