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    Unique Factorization Domain with Quotient Field

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    Let R be an integral domain with quotient field F and let
    p (X) be a monic polynomial in R[X] : Assume that p (X) = a (X) b (X)
    where a (X) and b (X) are monic polynomials in F [X] of smaller degree
    than p (X) : Prove that if a (X)is not in R[X] then R is not a UFD(unique factorization domain). Deduce that Z[2sqrt2] is not a UFD.

    © BrainMass Inc. brainmass.com June 3, 2020, 7:39 pm ad1c9bdddf
    https://brainmass.com/math/integrals/unique-factorization-domain-quotient-field-107075

    Solution Preview

    R is not a UFD since Gauss' lemma does not hold; that is, if R is a UFD with field of fractions F, then every polynomial which is reducible in F[x] ...

    Solution Summary

    This is a proof regarding a unique factorization domain.

    $2.19

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