Let R be a UFD and let F be a field of fractions for R. If...

Let R be a UFD and let F be a field of fractions for R. If f(α) = 0, where f ∈ R [x] is monic and α ∈ F, show that α ∈ R

NOTE: A corollary is the fact that m ∈ Z and m is not an n^th power in Z, then ⁿ√m is irrational.

Expert Solution

An integral domain R with 1 is called a Unique Factorization Domain if every non-zero non-unit element in R can be factorized as a finite product of irreducibles and this factorization is unique up to multiplication by units. Observe that since R is a U.F.D., hence we can talk about g.c.d. and relative-primeness . The notion of g.c.d. is the essential ingredient for completing the proof.

Colby Messinger answered 6 months ago

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