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Re: Field extension  dimension splitting field.
Posted:
Jul 18, 2013 6:14 AM


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.
The two 2dimensional extensions k and Q(w) are linearly disjoint.
 Timothy Murphy email: gayleard /at/ eircom.net tel: +353862336090, +35312842366 School of Mathematics, Trinity College, Dublin 2, Ireland



