Results 1 to 2 of 2

Math Help - Splitting Fields

  1. #1
    Junior Member
    Joined
    Apr 2010
    Posts
    58

    Splitting Fields

    Determine the splitting fields in C for the polynomials (over Q).
    a) x^3 - 1
    b) x^4 - 1
    c) x^3 + 3x^2 + 3x - 4

    a) x^3 = 1
    x = 1
    also, x^3 - 1 = (x - 1)(x^2 + x + 1)
    w = (-1 + √-3)/2 = -1/2 + (√3)/2i, where w is a root of x^2 + x + 1
    Q(1,w)

    b) x^4 = 1
    x = 1
    also, x^4 - 1 = (x^2 - 1)(x^2 + 1)
    Q(1,√i)

    c) I am not sure about.

    Can anyone let me know if my answer for a & b are correct and if not give me some help. I also need help with c.

    Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by page929 View Post
    Determine the splitting fields in C for the polynomials (over Q).
    a) x^3 - 1
    b) x^4 - 1
    c) x^3 + 3x^2 + 3x - 4

    a) x^3 = 1
    x = 1
    also, x^3 - 1 = (x - 1)(x^2 + x + 1)
    w = (-1 + √-3)/2 = -1/2 + (√3)/2i, where w is a root of x^2 + x + 1
    Q(1,w)


    Correct, but in fact \mathbb{Q}(1,w)=\mathbb{Q}(w)\,,\,w^3=1\,,\,w\neq 1


    b) x^4 = 1
    x = 1
    also, x^4 - 1 = (x^2 - 1)(x^2 + 1)
    Q(1,√i)


    This is incorrect: since both roots of x^2-1 are rational, we only need the roots of x^2+1 ,

    which are \pm i=\pm \sqrt{-1} , so here the splitting field is \mathbb{Q}(i)



    c) I am not sure about.


    Using basic calculus it's easy to see that this pol. has a real non-rational root between

    0 and 1 and, since it is a monotone ascending function of x, that one is the only real root, so here

    the splitting field is \mathbb{Q}(r,w) , with r the real root and w one of the two conjugate

    complex non-real roots.

    Tonio


    Can anyone let me know if my answer for a & b are correct and if not give me some help. I also need help with c.

    Thanks in advance.
    .
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Splitting fields
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 22nd 2011, 10:32 PM
  2. splitting fields
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: April 5th 2010, 05:11 AM
  3. Splitting fields
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: March 8th 2010, 08:33 AM
  4. Splitting fields
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: November 19th 2009, 07:40 AM
  5. Extension fields / splitting fields proof...
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: December 19th 2007, 07:29 AM

Search Tags


/mathhelpforum @mathhelpforum