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Math Help - field

  1. #1
    Junior Member
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    field

    help needed , I have found this one in a book but it has no answer.
    how can I show for any finite field F_q of even characterestic , the ring F_q[x] /(x^9+x^5+x^3+x+1) cannot be a field?
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  2. #2
    Senior Member roninpro's Avatar
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    Just to clarify: the characteristic of a field is either zero or a prime number, so "even characteristic" really just means characteristic 2.

    Recall that F_q[x]/(p(x)) is a field if and only if p is irreducible. However, the polynomial in question is reducible: x^9+x^5+x^3+x+1=\left(1+x+x^2\right) \left(1+x^2+x^4+x^6+x^7\right), so F_q[x] /(x^9+x^5+x^3+x+1) cannot be a field.
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