1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Arithmetic on polynomials

  1. May 16, 2015 #1
    1. The problem statement, all variables and given/known data
    What is the greatest common divisor of ##X^a - 1 ## and ## X^b - 1 ##, ##(a,b) \in \mathbb{N}^\star## ?

    2. Relevant equations


    3. The attempt at a solution

    Assuming that ## a\le b ##, I find by euclidian division of ##b## by ##a## that

    ## b = an + r \Rightarrow X^b - 1 = (X^a-1) (\sum_{k=1}^{n} X ^ {b-ka}) + X ^r - 1 ##

    So ##\text{gcd}(X^b - 1,X^a - 1) = \text{gcd}(X^a - 1,X^r - 1) ##
    So if I apply Euclid's algorithm on ##a## and ##b##, I will automatically get that the last non-zero remainder, which is ## \text{gcd}(a,b)##, guarantees that ## X ^{ \text{gcd}(a,b) } - 1 ## is the last non-zero remainder of the algorithm applied on ##X^a - 1 ## and ## X^b - 1 ##, and therefore is their greatest common divisor. Right ?
     
  2. jcsd
  3. May 16, 2015 #2
    Yes, It is the greatest commun divisor :)
     
  4. May 16, 2015 #3
    OK thank you ;-)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Arithmetic on polynomials
  1. Modular Arithmetic (Replies: 23)

  2. Modular arithmetic (Replies: 3)

  3. Arithmetics problem (Replies: 5)

  4. Cardinal arithmetic (Replies: 39)

  5. Arithmetic homework (Replies: 1)

Loading...