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Homework Help: Gcd(a,b) unique in Euclidean domain?

  1. Apr 9, 2012 #1
    gcd(a,b) unique in Euclidean domain??

    1. The problem statement, all variables and given/known data
    In Hungerford's Algebra on page 142, the problem 13 describes Euclidean algorithm on a Euclidean domain R to find THE greatest common divisor of a,b in R.

    My question is that does this THE mean THE UNIUQE? I've heard from my lecturer in a general commutative ring, a greatest common divisor of a,b in R does not have to be unique.

    Is there any theorem such as states that if R is a Euclidean domain, then for any a,b in R, gcd(a,b) is unique?

    2. Relevant equations

    3. The attempt at a solution
    Sorry, I have not figured out at all...
  2. jcsd
  3. Apr 9, 2012 #2


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    Re: gcd(a,b) unique in Euclidean domain??

    It's not even unique in the integers: 5 and -5 are both greatest common divisors of 20 and 35, for example. However, there is a simple relationship between all of the possibilities for a gcd....
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