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Finding a splitting field over Q

  1. Oct 28, 2016 #1
    1. The problem statement, all variables and given/known data
    I need to construct a splitting field of f(x)=x^4-x^3-5x+5 over Q

    2. Relevant equations


    3. The attempt at a solution
    So first I will assume r is a root and divide f(x) by (x-r). The quotient came out to be x^3 + (r-1)x^2 + (r^2-r)x + r^3 - r^2 - 5. I am a bit confused what to do now, do i assume another root, call it m perhaps, and do the division again?
     
  2. jcsd
  3. Oct 28, 2016 #2

    pasmith

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    Homework Helper

    Always check first for obvious integer roots.

    Most obvious is: 0 is a root if and only if there is no constant term.

    Next most obvious is: 1 is a root if and only if the sum of the coefficients is zero.
     
    Last edited: Oct 28, 2016
  4. Oct 30, 2016 #3
    Wow, I didn't know that condition for 1 being a root, thanks! I have now reduced my expression to (x-1)(x^3-5). Will the roots of x^3-5 be the cube root of 5, and then the product of the cube root of 5 with the first and second powers of a third root of unity?
     
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