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Finding the splitting field of x^4-7x in C over Q

  1. Oct 22, 2016 #1
    1. The problem statement, all variables and given/known data
    Hello PF. I need to find a splitting field of x^4-7x in C over Q

    2. Relevant equations


    3. The attempt at a solution
    letting r be a root, I did the division and got x^4-7x = (x-r)(x^3+r*x^2+x*r^2+r^3). I'm a little confused on what to do now, do I just take another root and do the division again?
     
  2. jcsd
  3. Oct 22, 2016 #2

    mfb

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    2016 Award

    Staff: Mentor

    You can find all the roots (in C) and see how many of them are in Q.
     
  4. Oct 22, 2016 #3
    Is the best way to find all the roots in C to do what i've been doing? Assume an element is a root and then divide?
     
  5. Oct 22, 2016 #4

    pasmith

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    Homework Helper

    The right hand side is [itex]x^4 - r^4[/itex] which for fixed [itex]r[/itex] is not identically equal to the left hand side for every [itex]x[/itex].

    Starting with [itex]x^4 - 7x = (x - r)(x^3 + ax^2 + bx + c)[/itex] and comparing coefficients of powers of [itex]x[/itex] leads to [tex]
    a - r = 0, \\
    b - ar = 0, \\
    c - br = -7, \\
    cr = 0.[/tex] This is a system in four unknowns [itex]a[/itex], [itex]b[/itex], [itex]c[/itex] and [itex]r[/itex] which has the solution [itex]a = r[/itex], [itex]b = r^2[/itex], [itex]c = r^3 - 7[/itex] and [itex]r(r^3 - 7) = 0[/itex]. This of course gets you no closer to actually finding [itex]r[/itex].

    Instead observe that [itex]x^4 - 7x = x(x^3 - 7)[/itex] and then use the identity [itex]x^3 - r^3 \equiv (x - r)(x^2 + rx + r^2)[/itex]. That leaves you to factorize a quadratic.
     
  6. Oct 22, 2016 #5
    Where r = (7)^1/3? Thank you by the way. I feel like this was a really obvious question in retrospect
     
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