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Math Help - polynomial

  1. #1
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    polynomial

    For each divisor m of 6, give an example of a polynomial p(x) in Q[x], such that the splitting field E of p(x) over Q has m, ie [E:Q]=m
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  2. #2
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    for m=3 you have to use some cubic polynomial whose determinant is a square in Q, so that the Galois group does not contain any odd permutations
    for other cases, cyclotomic polynomial suffices
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  3. #3
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    Quote Originally Posted by FancyMouse View Post
    for m=3 you have to use some cubic polynomial whose determinant is a square in Q, so that the Galois group does not contain any odd permutations
    for other cases, cyclotomic polynomial suffices
    Seems like a slight typo there. It should be a "discriminant".

    For a concrete example,

    If ch(F) \neq 2, then the permutation \sigma \in S_n is an element of A_n if and only if it fixes the square root of the discriminant D.

    We know that A_3 = Z_3. So the possible polynomial can be

    f(x) = x^3 + x^2 - 2x -1  \in \mathbb{Q}[x] and its discriminant is 49. Then, if the splitting field of f(x) is K, [K:Q]=3 and the order of Gal(K/Q) is 3.
    Last edited by aliceinwonderland; April 27th 2010 at 10:37 PM.
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  4. #4
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    >"discriminant".
    Ah yes, thanks for pointing out.
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