# Splitting field

• Sep 2nd 2009, 11:07 PM
MAX09
Splitting field
I was working on a problem to find the degree of the splitting field of the polynomial x^3-11 over Q

I figured out to some extent. x^3-11 =0 has roots r*1,r*w,r*w^2 as roots where r is the cubic root of 11; w and w^2 are the cubic roots of unity.
I realize that the w cannot belong to Q as it is complex. My doubt is with the degree of the splitting field. I've got the answer for the degree to be 6.

Can anyone give a brief explanation of how it is 6?

Thanks,

MAX
• Sep 2nd 2009, 11:27 PM
ynj
Quote:

Originally Posted by MAX09
I was working on a problem to find the degree of the splitting field of the polynomial x^3-11 over Q

I figured out to some extent. x^3-11 =0 has roots r*1,r*w,r*w^2 as roots where r is the cubic root of 11; w and w^2 are the cubic roots of unity.
I realize that the w cannot belong to Q as it is complex. My doubt is with the degree of the splitting field. I've got the answer for the degree to be 6.

Can anyone give a brief explanation of how it is 6?

Thanks,

MAX

The splitting field is $\displaystyle Q(r\omega,r)$.
$\displaystyle [Q(r\omega,r):Q]=[Q(r\omega,r):Q(r)][Q(r):Q]=6$,since$\displaystyle r^3=11,(r\omega)^2+r\cdot r\omega+r^2=0$
• Sep 2nd 2009, 11:33 PM
MAX09
:)
Thanks for the help, ynj. reply was exactly the same I was looking for.

Thanks a bunch :)