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Math Help - Roots of Unity

  1. #1
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    Roots of Unity

    Given t^3 -1 = 0, where t is an element of an algebraically closed field, how many distinct roots of unity must this equation have? If I'm working over the complex numbers, of course I'll have 3 distinct roots, but is there a theorem of something that says t^3 -1 =0 has 3 distinct roots in any algebraically closed field?
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  2. #2
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    Re: Roots of Unity

    Never mind I got it. A polynomial is separable if all of its roots are distinct, and f has a multiple root x iff x is also a root of D_x(f(x)). So 0 is a root of 3t^2, but not a root of t^3-1, so t^3-1 is separable (i.e. has 3 distinct roots).
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