I am working on Exercise 8 of Dummit and Foote Section 9.2 Exercise 8

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Determine the greatest common divisor of $\displaystyle a(x) = x^3 - 2 $ and $\displaystyle b(x) = x + 1 $ in $\displaystyle \mathbb{Q} [x] $

and write it as a linear combination (in $\displaystyle \mathbb{Q} [x] $ ) of a(x) and b(x).

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In working on this I applied the Division Algorithm to a(x) and b(x) resulting in

$\displaystyle x^3 - 2 = (x^2 - x + 1) (x+ 1) + (-3) $

then

$\displaystyle (x + 1) = (1/3 x + 1/3) + 0 $

Last non-zero remainder is -3

Therefore, gcd is -3

BUT!

This does not seem to be correct because -3 does not divide either a(x) and b(x)

Can someone please help?

Peter