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September 10th, 2012, 08:15 AM   #1
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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
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September 10th, 2012, 11:33 AM   #2
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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.

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October 19th, 2012, 06:38 AM   #3
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Re: unique factorization domain, zero of a polynomial , proo

thanks, I finally understand it
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