Greatest common divisor of polynomials

[evinda]

Hello!

I want to find the greatest common divisor of $x^4+1$ and $x^2+x+1$.
I applied the Euclidean division and found that $x^4+1=(x^2+x+1) \cdot (x^2-x)+(x+1)$.
So,isn't it like that: $gcd(x^4+1,x^2+x+1)=x+1$ ?
But.. in my textbook,the result is $1$! Which of the both results is right?? (Blush) (Thinking)

 

[Evgeny.Makarov]

You did not finish the Euclidean algorithm, which consists of several Euclidean divisions.

 

[evinda]

You did not finish the Euclidean algorithm, which consists of several Euclidean divisions.

Oh,yes!You are right! (Giggle) I found now that the greatest common divisor is equal to $1$! Thank you! :)

 

[chisigma]

Hello!

I want to find the greatest common divisor of $x^4+1$ and $x^2+x+1$.
I applied the Euclidean division and found that $x^4+1=(x^2+x+1) \cdot (x^2-x)+(x+1)$.
So,isn't it like that: $gcd(x^4+1,x^2+x+1)=x+1$ ?
But.. in my textbook,the result is $1$! Which of the both results is right?? (Blush) (Thinking)

If they are 'ordinary polynomials' then their factorisations are...

$\displaystyle 1 + x^{4} = (x - e^{i\ \frac{\pi}{4}})\ (x - e^{- i\ \frac{\pi}{4}})\ (x - e^{i\ \frac{3\ \pi}{4}})\ (x - e^{- i\ \frac{3\ \pi}{4}})$

$\displaystyle 1 + x + x^{2} = (x - e^{i\ \frac{2\ \pi}{3}})\ (x - e^{- i\ \frac{2\ \pi}{3}})$

... and the two polynomial have no common factors so that the g.c.d. is 1.

If they are 'integer modulo 2 polynomials' however then their factorisations are...$\displaystyle 1 + x^{4} = (1 + x)\ (1 + x)\ (1 + x + x^{2})$$\displaystyle 1 + x + x^{2} = 1 + x + x^{2}$ ... so that the g.c.d. is $\displaystyle 1 + x + x^{2}$...

Kind regards $\chi$ $\sigma$

 

[blue_raver22]

Hello! It appears that you have correctly calculated the greatest common divisor using Euclidean division. However, it is possible that the textbook is referring to the greatest common divisor in the context of polynomials over a specific field or ring. In this case, the greatest common divisor would be the monic polynomial of highest degree that divides both $x^4+1$ and $x^2+x+1$. Since both polynomials have a degree of 4, the monic polynomial of highest degree would be 1, hence the result given in the textbook. Both results are correct, it just depends on the context in which you are considering the greatest common divisor. I hope this helps clarify any confusion!