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Math Help - identify galois group

  1. #1
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    Question identify galois group

    f= t^3 -6t +6 in Q[t]

    basically my first part of the question was to find the zeros of f, i have done that part already but the part i got stuck on was that i want to identify the galois group of f, but not sure how to, so can anyone explain why it is that galois group thanks
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  2. #2
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    Quote Originally Posted by dopi View Post
    f= t^3 -6t +6 in Q[t]

    basically my first part of the question was to find the zeros of f, i have done that part already but the part i got stuck on was that i want to identify the galois group of f, but not sure how to, so can anyone explain why it is that galois group thanks
    This irreducible polynomial has 1 real root and 2 complex roots that are complex conjugates. If \tau is complex conjugation then we see that \tau is a transposition (2-cycle) if viewed as an element of S_3 (remember that the Galois group is a subgroup of S_3). Now if \alpha is a root of f then \mathbb{Q}(\alpha)/\mathbb{Q} is a degree 3 extension and so 3 divides the degree of the splitting field over \mathbb{Q}. Thus, 3 divides the order of the Galois group, hence there is an element of order 3 in the Galois group i.e. a 3-cycle. But any subgroup of S_3 generated by a 2-cycle and a 3-cycle is S_3 itself, so the Galois group is S_3.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    This irreducible polynomial has 1 real root and 2 complex roots that are complex conjugates. If \tau is complex conjugation then we see that \tau is a transposition (2-cycle) if viewed as an element of S_3 (remember that the Galois group is a subgroup of S_3). Now if \alpha is a root of f then \mathbb{Q}(\alpha)/\mathbb{Q} is a degree 3 extension and so 3 divides the degree of the splitting field over \mathbb{Q}. Thus, 3 divides the order of the Galois group, hence there is an element of order 3 in the Galois group i.e. a 3-cycle. But any subgroup of S_3 generated by a 2-cycle and a 3-cycle is S_3 itself, so the Galois group is S_3.
    awsome thank you
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  4. #4
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    You could also find the discriminant of the polynomail, which is -108 and since it is not a square in Q(rationals) and the polynomail is irreducible 6-Eisenstein, you have that the Galois group is S_3.
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