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Math Help - A simple problem about prime and irreducible

  1. #1
    ynj
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    A simple problem about prime and irreducible

    Let D be a UFD. a,b\in D, if gcd(a,b)is not a unit, will there always exists a prime pthat is the common divisor of a,b?why?
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    Quote Originally Posted by ynj View Post
    Let D be a UFD. a,b\in D, if gcd(a,b)is not a unit, will there always exists a prime pthat is the common divisor of a,b?why?
    of course. just choose p to be any prime element in the prime factorization of \gcd(a,b).
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  3. #3
    ynj
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    prime factoralization? but the definition of UFD says that any nonunit a can be factorlized in to product of irreducible elements, say a=p_1...p_n, where p_1...p_n. Do you mean that p_iis prime,or there is other theorem named"prime factoralization"?
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  4. #4
    ynj
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    oh,sorry, i have just searched the internet, and find a proof which says that irreducible and prime are equivalent in UFD
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