I understand that there are other posts on the forum about the same topic, but, after reading them, I didn't understand exactly what to do in this situation.
What I've done so far. In this same exercise, I have already shown that $\Bbb Z[x]$ is an UFD (obvious, since $\Bbb Z$ is an UFD and using Gauss' Theorem) but not an euclidian domain (to do this, I considered the ideal $(2,x)$ and showed this isn't a principal ideal and thus $\Bbb Z[x]$ isn't a PID $\Rightarrow$ $\Bbb Z[x]$ is not an Euclidian domain). Besides from this, I was also asked to decompose the polynomial $1+x+x^2+x^3$ in irreducibles and I came up with the following conclusion: \begin{equation*} f(x) = 1 + x + x^2 + x^3 = (1+x)(1+x^2) \end{equation*} which are both irreducible polynomials over $\Bbb Z[x]$. But after this, I am asked to compute the $\gcd$ between $f$ and $f'$ and I don't know how to proceed with $\gcd$ calculation over a NOT Euclidian domain, like $\Bbb Z$. Is my decomposition in irreducibles helpfull somehow?
Example. Altough there is no algorithm in $\Bbb Z[x]$ based on an euclidian evaluation to compute $\gcd$'s, compute the following: \begin{equation*} \gcd(1+x+x^2+x^3,1+2x+3x^2) \end{equation*}
Thanks for any help in advance.
Here is one way to approach the problem.
Let your polynomials be $P_1(x)=1+x+x^2+x^3$ and $P_2(x)=1+2x+3x^2$. Any common factor of $P_1$ and $P_2$ will also be a common factor of any linear combination of $P_1$ and $P_2$ (linear combination with coefficients coming from $\mathbb{Z}[x]$). Also linear combinations of linear combinations of $P_1$ and $P_2$ will again be linear combinations of $P_1$ and $P_2$.
So any any common factor of $P_1$ and $P_2$ will also be a factor of $P_3=3P_1-xP_2=3+2x+x^2$.
Then any common factor of $P_1$ and $P_2$ will also be a factor of $P_4=3P_3-P_2=8+4x$.
But $P_4(x)$ has irreducible factors of $2$ and $2+x$, neither of which is a factor of $P_1$ (or of $P_2$ for that matter), so $P_1$ and $P_2$ have no common factors other than $1$ (and $-1$).