<s>[INST] Determine the greatest common divisor of polynomials $x^2+1$ and $x^3+1$ in $\Bbb Q[X]$. Exercise: Determine a gcd of the polynomials $x^2+1$ and $x^3+1$ in $\Bbb Q[X]$.. Write the gcd as a combination of the given polynomials.
Is it correct that I keep using long division until the result is $0$, and then the previous result would be a gcd?
x^2+1 / x^3 + 1 \ x
        x^3 + x
        ________-
        1 - x

1 - x / x^2 + 1 \ -x-1
        x^2 - x 
        ________-
        x + 1
        x - 1
        ________-
        2

2 / 1 - x \ -1/2x + 1/2 
      - x 
    ______-
    1
    1
    ______-
    0

Conclusion: A gcd is $2$?
I'm not sure what's meant with "Write the gcd as a combination of the given polynomials.", so it would be great if someone could point me in the right direction.
 [/INST] $$  \left(   x^{3}  + 1 \right)  $$ 
$$  \left(   x^{2}  + 1 \right)  $$ 
$$  \left(   x^{3}  + 1 \right)  =  \left(   x^{2}  + 1 \right)  \cdot \color{magenta}{  \left(   x  \right) } +  \left(   -  x  + 1 \right)  $$ 
 $$  \left(   x^{2}  + 1 \right)  =  \left(   -  x  + 1 \right)  \cdot \color{magenta}{  \left(   -  x  - 1 \right) } +  \left( 2  \right)  $$ 
 $$  \left(   -  x  + 1 \right)  =  \left( 2  \right)  \cdot \color{magenta}{  \left(   \frac{  -  x  + 1 }{ 2 }  \right) } +  \left( 0 \right)  $$ 
 $$ \frac{ 0}{1} $$ 
 $$ \frac{ 1}{0} $$ 
 $$ \color{magenta}{  \left(   x  \right) }  \Longrightarrow  \Longrightarrow  \frac{  \left(   x  \right) }{ \left( 1  \right) } $$ 
 $$ \color{magenta}{  \left(   -  x  - 1 \right) }  \Longrightarrow  \Longrightarrow  \frac{  \left(   -  x^{2}  -  x  + 1 \right) }{ \left(   -  x  - 1 \right) } $$ 
 $$ \color{magenta}{  \left(   \frac{  -  x  + 1 }{ 2 }  \right) }  \Longrightarrow  \Longrightarrow  \frac{  \left(   \frac{  x^{3}  + 1 }{ 2 }  \right) }{ \left(   \frac{  x^{2}  + 1 }{ 2 }  \right) } $$ 
 $$  \left(   x^{3}  + 1 \right)  \left(   \frac{  -  x  - 1 }{ 2 }  \right)  -  \left(   x^{2}  + 1 \right)  \left(   \frac{  -  x^{2}  -  x  + 1 }{ 2 }  \right)  =  \left( -1  \right)  $$ 
..................................
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