Explore BrainMass

Explore BrainMass

    Unique Factorization Domain with Quotient Field

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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

    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.