Gcd(a,b) unique in Euclidean domain?

In summary, the conversation is about the uniqueness of the greatest common divisor (gcd) in a Euclidean domain. The speaker is questioning whether the use of "THE" in the problem description means that the gcd is unique. They mention hearing from a lecturer that in a general commutative ring, the gcd does not have to be unique. The other person asks if there is a theorem stating that in a Euclidean domain, the gcd is always unique. The attempt at a solution mentions that the gcd is not always unique, even in the integers, but there is a simple relationship between all the possible values for a gcd.
  • #1
julypraise
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gcd(a,b) unique in Euclidean domain??

Homework Statement


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?

Homework Equations


The Attempt at a Solution


Sorry, I have not figured out at all...
 
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  • #2


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...
 

Related to Gcd(a,b) unique in Euclidean domain?

1. What is a Euclidean domain?

A Euclidean domain is a mathematical structure that consists of a set of elements and two operations (usually addition and multiplication) that satisfy certain properties. In a Euclidean domain, you can perform division with remainder, meaning that for any two elements a and b, there exists a quotient q and a remainder r such that a = bq + r, where r is either zero or has a smaller "size" than b.

2. What is the greatest common divisor (GCD) in a Euclidean domain?

The greatest common divisor (GCD) of two elements a and b in a Euclidean domain is the largest element that divides both a and b without leaving a remainder. In other words, it is the largest element that is a common divisor of a and b.

3. Why is GCD(a,b) unique in a Euclidean domain?

In a Euclidean domain, the GCD of two elements a and b is unique because it is defined as the largest common divisor of a and b. This means that there cannot be any other element that is larger and also divides both a and b without leaving a remainder.

4. How is GCD(a,b) calculated in a Euclidean domain?

The GCD of two elements a and b in a Euclidean domain can be calculated using the Euclidean algorithm, which involves repeatedly dividing the larger element by the smaller element and using the remainder as the new divisor. This process is repeated until the remainder is zero, at which point the last non-zero remainder is the GCD of a and b.

5. What is the significance of GCD(a,b) in a Euclidean domain?

The GCD of two elements a and b in a Euclidean domain has many important applications, such as simplifying fractions, finding the lowest common denominator, and solving linear Diophantine equations. It is also used in many other areas of mathematics, such as cryptography and number theory.

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