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September 10th, 2012, 08:15 AM  #1 
Senior Member Joined: Sep 2010 From: Germany Posts: 153 Thanks: 0  unique factorization domain, zero of a polynomial , proof
let A be a UFD and K its field of fractions. and where is a monic polynomial. Prove that if f has a root , then in fact I need some guidance with the proof. Proof: which gives and we observe that *since all the terms on the rhs are multiples of d but how do we know that ??? from what do we conclude that And later in the proof it says '' hence c,d are relatively prime we get that how do we deduce the last part as well? Any help appreciated! My questions are: how do we know that how do we know that c, d are relatively prime? based on what do we conclude that 
September 10th, 2012, 11:33 AM  #2 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: unique factorization domain, zero of a polynomial , proo
If c and d are not relatively prime then they have a common factor p which can be divided out at any time from so it suffices to consider c and d relatively prime. now won't divide unless because c and d are relatively prime. 
October 19th, 2012, 06:38 AM  #3 
Senior Member Joined: Sep 2010 From: Germany Posts: 153 Thanks: 0  Re: unique factorization domain, zero of a polynomial , proo
thanks, I finally understand it 

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domain, factorization, polynomial, proof, unique 
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