So I have the polynomials $f(x) = x^3+x^2+x$ and $g(x)=x^2+x+1$

We are told to find the $gcd$ of both these polynomials in $Q[x], Z/3Z[x], Z/5Z[x], Z/11Z[x]$

After applying the Euclidean algorithm, I see that:

$x^3+x^2+x = x(x^2+x+1) + 0$

Thus the gcd is $x^2+x+1$

However, when I attempt to do the same in all the other fields, I end up with the same answer. I am doing this incorrectly or is it actually possible to have the same gcd in all fields?

  • 2
    $\begingroup$ Are you certain that you have the correct polynomials? Also, note that the $\gcd$ of $f$ and $f$ is $f$ over any field, for any $f$, so something like that is definitely possible. $\endgroup$ – Arthur Oct 17 at 5:25
  • $\begingroup$ Yes, the polynomials are correct. $\endgroup$ – flutterbug98 Oct 17 at 5:27
  • 1
    $\begingroup$ If those are really the right polynomials, then it's a silly question. No need to use the Euclidean algorithm: $g$ divides $f$, so obviously their GCD is just $g$. $\endgroup$ – J Swanson Oct 17 at 5:48
  • $\begingroup$ Okay, thanks for the help. To be honest, it did seem a bit weird to me as well! $\endgroup$ – flutterbug98 Oct 17 at 5:55

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