Different answers for `GCD` of 2 polynomials. Wolfram answer differs from answer obtained from my program of Euclids algorithm.

Question

The GCD of two polynomials

$f_1= (x^6 - 1)$,

$f2= (x^4 + 2x^3 + 2x^2 − 2x − 3)$.

as per Wolfram alpha is $(x^2-1)$. But as per my Euclidean algorithm program the GCD is $(57/4)x^2 - (57/4)$.

The answer I got divides both the polynomials exactly and is greater than the answer of Wolfram alpha. So, which answer is the correct GCD?

Answer 1

It turns out that the greatest common divisor isn't actually unique--it's only unique up to multiplication by a unit.

So in the ring $\mathbb{Q}[x]$, both of those are the greatest common divisors, because you can multiply either by an invertible element to get the other one.

Now, on the other hand, sometimes people specify that the greatest common divisor be monic. That depends on how you've defined it.