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Math Help - Finite fields....

  1. #1
    AAM
    AAM is offline
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    Finite fields....

    Hi everyone,

    I'm trouble solving the following question. (note - calculator not allowed)

    Prove that this only 1 distinct intermediate subgroup between K = finite field of order 2 adjoined with Theta a root of X^4+X^3+1 & the finite field of order 2.

    I attempted to show that the elements fixed by sigma, sigma^2 & sigma^3 are the same (where sigma is the Frobenius map A |---> A^2 ), but I just get lost in calculation! :-s

    Is there an easier method?

    Many thanks. x
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  2. #2
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    Quote Originally Posted by AAM View Post
    Hi everyone,

    I'm trouble solving the following question. (note - calculator not allowed)

    Prove that this only 1 distinct intermediate subgroup between K = finite field of order 2 adjoined with Theta a root of X^4+X^3+1 & the finite field of order 2.

    I attempted to show that the elements fixed by sigma, sigma^2 & sigma^3 are the same (where sigma is the Frobenius map A |---> A^2 ), but I just get lost in calculation! :-s

    Is there an easier method?

    Many thanks. x
    The polynomial x^4+x^3+1 is irreducible over \mathbb{F}_2. Let \theta be a root in some larger extension field. Construct K = \mathbb{F}_2(\theta). Now [K:F] = 4 and \text{Gal}(K/F) = \left< \sigma \right> where \sigma is the Frobenius automorphism i.e. \sigma: K\to K by \sigma (x) = x^2. Therefore, the Galois group is cyclic. The intermediate subfields of degree 2 over \mathbb{F}_2 correspond to subgroup of the Galois group of index 2. However, \text{Gal}(K/F) \simeq \mathbb{Z}_4 and so there is only one subgroup of index 2.
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