Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.independent

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Nick

Posts: 20
Registered: 4/12/08
Re: Field extension - dimension splitting field.
Posted: Jul 18, 2013 10:32 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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




Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.