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

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 Nick Posts: 50 Registered: 4/12/08
Re: Field extension - dimension splitting field.
Posted: Jul 18, 2013 10:32 AM

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

Date Subject Author
7/18/13 Nick
7/18/13 Timothy Murphy
7/18/13 Nick
7/18/13 Peter Percival
7/18/13 Nick
7/18/13 Peter Percival
7/18/13 fom