Vtyjkyuk

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.

The Euclidean algorithm steps are
beginalign*
x^3+1&=x (x^2+1) + (-x+1)\
x^2+1&=-x (-x+1)+ (x+1)\
-x+1&=-(x+1)+2
endalign*

So the GCD is $1$.

We can go back up
beginalign*
2 &= (-x+1) +(x+1)\
&= (-x+1) + (x^2+1)+x (-x+1)\
&= (-x+1) (x+1)+(x^2+1)\
&= (x^3+1 - x(x^2+1))(x+1)+(x^2+1)\
&= (x+1) (x^3+1) + (-x^2-x+1) (x^2+1)
endalign*

and so $1 = frac12 [(x+1) (x^3+1) + (-x^2-x+1) (x^2+1)]$

Notice that $x^2+1$ is irreducible over $mathbbQ$. Let $d = gcd(x^2+1, x^3+1)$. Then $d mid x^2+1$ so $d$ is a nonzero constant, or $d$ is an associate of $x^2+1$.

However, $x^2+1 notmid x^3+1$ so $d$ must be constant. We can choose $d = 1$ since $gcd$ is usually supposed to be monic.

Finally, notice that

$$1=left[frac12(x+1) right](x^3+1) + left[frac12(-x^2-x+1)right] (x^2+1)$$

$$ left( x^3 + 1 right) $$

$$ left( x^2 + 1 right) $$

$$ left( x^3 + 1 right) = left( x^2 + 1 right) cdot colormagenta left( x right) + left( - x + 1 right) $$
$$ left( x^2 + 1 right) = left( - x + 1 right) cdot colormagenta left( - x - 1 right) + left( 2 right) $$
$$ left( - x + 1 right) = left( 2 right) cdot colormagenta left( frac - x + 1 2 right) + left( 0 right) $$
$$ frac 01 $$
$$ frac 10 $$
$$ colormagenta left( x right) Longrightarrow Longrightarrow frac left( x right) left( 1 right) $$
$$ colormagenta left( - x - 1 right) Longrightarrow Longrightarrow frac left( - x^2 - x + 1 right) left( - x - 1 right) $$
$$ colormagenta left( frac - x + 1 2 right) Longrightarrow Longrightarrow frac left( frac x^3 + 1 2 right) left( frac x^2 + 1 2 right) $$
$$ left( x^3 + 1 right) left( frac - x - 1 2 right) - left( x^2 + 1 right) left( frac - x^2 - x + 1 2 right) = left( -1 right) $$

..................................