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Math Help - Splitting field

  1. #1
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    Lightbulb 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
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  2. #2
    ynj
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    Quote Originally Posted by MAX09 View Post
    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 Q(r\omega,r).
    [Q(r\omega,r):Q]=[Q(r\omega,r):Q(r)][Q(r):Q]=6,since r^3=11,(r\omega)^2+r\cdot r\omega+r^2=0
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  3. #3
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    Thumbs up :)

    Thanks for the help, ynj. reply was exactly the same I was looking for.

    Thanks a bunch
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