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November 18th, 2006, 05:54 PM   #1
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Application of Gauss' lemma

"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) does not belong to R[x] then R is not a Unique Factorization Domain."

By Gauss' lemma, if R is a UFD, then a and b must lie in R[X], which proves the result.

"Deduce that Z[2*sqrt2] is not a U.F.D"

Consider (X-sqrt(2)/2)(X+sqrt(2)/2).
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