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Topic: Field extension - dimension splitting field.
Replies: 6   Last Post: Jul 18, 2013 11:42 AM

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Timothy Murphy

Posts: 55
Registered: 12/27/11
Re: Field extension - dimension splitting field.
Posted: Jul 18, 2013 6:14 AM
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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 2-dimensional extensions k and Q(w) are linearly disjoint.

Timothy Murphy
e-mail: gayleard /at/
tel: +353-86-2336090, +353-1-2842366
School of Mathematics, Trinity College, Dublin 2, Ireland

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