On 18/07/2013 11:14, Timothy Murphy wrote: > Nick wrote: > >> >> An exercise in a book (Galois Theory, Ian Stewart) asks for the >> splitting field of polynomial t^6 - 8 over the Rationals and the degree >> of this splitting field. >> >> The book answer is that the splitting field is Q(2^(1/2), e^(i*pi/3)) >> and that its degree is 12. >> >> Now I think the splitting field given in the book is ok but that its >> degree is 4 not 12. >> >> My preferred representation of the splitting field extension would be >> Q(2^(1/2), i * 3^(1/2)). >> >> Unfortunately the book is from 1982 and there doesn't appear to be >> published errata for this edition on the web. >> >> >> I'm pretty sure I'm right but would like to check. > > I agree with your conclusion. > > If k = Q(sqrt2) then the splitting field is K = k(w), where w^2 + w + 1 = 0. > So K has dimension 2 x 2 = 4. >
I can see that w^2+w+1 is a valid minimum monic polynomial for the extension, its not the one I picked but I guess there are many valid choices.
> The two 2-dimensional extensions k and Q(w) are linearly disjoint. > > Thanks