Gcd(a,b) unique in Euclidean domain?

  • Thread starter Thread starter julypraise
  • Start date Start date
  • Tags Tags
    Domain Euclidean
Click For Summary
SUMMARY

The discussion centers on the uniqueness of the greatest common divisor (gcd) in a Euclidean domain, specifically referencing Hungerford's Algebra. The participant questions whether the term "THE" in the context of gcd implies uniqueness, noting that in general commutative rings, gcds are not necessarily unique. The consensus is that while gcds may not be unique in broader contexts, in a Euclidean domain, the gcd is indeed unique up to multiplication by a unit.

PREREQUISITES
  • Understanding of Euclidean domains and their properties
  • Familiarity with the concept of greatest common divisors (gcd)
  • Basic knowledge of commutative rings
  • Awareness of units in algebraic structures
NEXT STEPS
  • Study the properties of Euclidean domains in detail
  • Examine the definitions and examples of greatest common divisors
  • Learn about units in algebra and their role in gcd uniqueness
  • Explore the implications of gcd uniqueness in various algebraic structures
USEFUL FOR

Mathematics students, algebra enthusiasts, and educators seeking clarity on the properties of gcds in Euclidean domains and their implications in algebraic theory.

julypraise
Messages
104
Reaction score
0
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...
 
Physics news on Phys.org


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...
 

Similar threads

Finding GCD with Fibonacci: Base Case
  • · Replies 1 ·
Replies
1
Views
2K
What is the Greatest Common Divisor of Two Polynomials?
  • · Replies 3 ·
Replies
3
Views
2K
Does the GCD of a set divide the GCD of any subset?
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Equivalent definitons for primitive polynomials
  • · Replies 24 ·
Replies
24
Views
2K
Number Theory. If d=gcd(a,b,c) then d is a linear combination of a,b, and c
  • · Replies 1 ·
Replies
1
Views
3K
Proving the GCD Property: A Challenge for Algebra Students
  • · Replies 1 ·
Replies
1
Views
2K
Constructing Cones as Orbifolds: Fundamental Domain for an Identification
  • · Replies 4 ·
Replies
4
Views
1K
Finding the inverse of a polynomial in a field
  • · Replies 1 ·
Replies
1
Views
4K
Tricky abstract algebra problem
  • · Replies 20 ·
Replies
20
Views
3K
Find GCD of a and b: Express as Integer Combination
  • · Replies 2 ·
Replies
2
Views
3K