Greatest common divisor of two polynomials

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

Re: Greatest common divisor of two polynomials

Hi,

I think your plan of attack is exactly correct. Remember you are working in Q[x], not Z[x]. So in Q[x], -3 __does__ divide any polynomial; e.g. $\displaystyle x^3-2 =-3(-1/3x^3-2/3)$. That is -3 is a unit in Q[x].

Re: Greatest common divisor of two polynomials

Using the rational roots theorem, you can check that $\displaystyle a(x)$ has no rational roots. I pretty sure this means it has no (non-unit) factors in $\displaystyle \mathbb{Q}\[x\]$. This would make $\displaystyle a(x) \in \mathbb{Q}\[x\]$ irreducible. Therefor, I believe $\displaystyle gcd( a(x), b(x) ) = u$, any unit in $\displaystyle \mathbb{Q}\[x\]$, i.e. any (non-zero) constant polynomial. Since there ordering on $\displaystyle \mathbb{Q}\[x\]$ is only concerned with the polynomials degree, I don't think any constant polynomial would be called "greater" then any other.