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I am trying to understand a discussion of field extensions by Dummit and Foote: Abstract Algebra, page 512 ...
The relevant text is as follows:View attachment 4832When I first read the second paragraph, starting "Given any field $$F$$ and any polynomial $$p(x) \in F[x]$$ ... ... ... ", I thought that D&F were trying to construct a situation where $$p(x)$$ did not have a root in $$F$$ ... ... and then go on to construct an extension field $$K = F[x]/p(x)$$ that did indeed have a root of the polynomial $$p(x)$$ ... ...BUT ... ... D&F appear (to me, anyway) to try to ensure that $$p(x)$$ does not have a root in $$F$$ by stipulating or declaring that $$p(x)$$ must be irreducible ... at least I think this is their intention .. ...
BUT ... ... ? ... ... linear polynomials such as $$x - 7$$ are irreducible in $$F[x]$$ when $$F = \mathbb{Q}$$ ... but $$p(x) = x - 7$$ does have a root in $$\mathbb{Q}$$ ... that is no need for an extension field (except the 'extension' $$\mathbb{Q} = \mathbb{Q}$$) ...
So it seems to me I am not fully understanding D&F's discussion ...
Can someone please clarify D&F's discussion for me by explaining what they are really getting at ...
Peter
The relevant text is as follows:View attachment 4832When I first read the second paragraph, starting "Given any field $$F$$ and any polynomial $$p(x) \in F[x]$$ ... ... ... ", I thought that D&F were trying to construct a situation where $$p(x)$$ did not have a root in $$F$$ ... ... and then go on to construct an extension field $$K = F[x]/p(x)$$ that did indeed have a root of the polynomial $$p(x)$$ ... ...BUT ... ... D&F appear (to me, anyway) to try to ensure that $$p(x)$$ does not have a root in $$F$$ by stipulating or declaring that $$p(x)$$ must be irreducible ... at least I think this is their intention .. ...
BUT ... ... ? ... ... linear polynomials such as $$x - 7$$ are irreducible in $$F[x]$$ when $$F = \mathbb{Q}$$ ... but $$p(x) = x - 7$$ does have a root in $$\mathbb{Q}$$ ... that is no need for an extension field (except the 'extension' $$\mathbb{Q} = \mathbb{Q}$$) ...
So it seems to me I am not fully understanding D&F's discussion ...
Can someone please clarify D&F's discussion for me by explaining what they are really getting at ...
Peter