<s>[INST] Constructing a polynomial with integer coefficient's sharing roots with another two polynomials 
Let $i$ be a root of the equation $ x
^2 + 1 = 0 $and let $ω$ be a root of the
equation $x^2 + x + 1 = 0$. Construct a polynomial $f(x) = \sum_k a_k x^k$  with $a_k \in \mathbb{Z}$ such that $f(i+w)=0$. Source

After some long thought, I got a 'partial' construction for $f$: let $q(x) =x^2 +1$ and $p(x) = x^2 + x+1$, then:
$$ f(x) = q(x- \omega) + p(x- i)$$
However, the above doesn't necessarily satisfy the 'integer' coefficients criteria and may contain an imaginary part, so I modified the construction a bit to fix the second problem and got:
$$ f(x) = q(x- \omega) + p(x-i) + \overline{ p(x-i) + q( x - \omega)}$$
Problem: The above has no guarantee to have integer coefficients and the expression also has terms of $\overline{x}$ i.e: complex conjugate of $x$.
 [/INST] Multiplying $\,\alpha=\omega+i\,$ by $\,i\,$ and using that $\,i^2=-1\,$:
$$
\begin{align}
\begin{cases}
\alpha &= \omega+i
\\ i \,\alpha &= i \omega - 1
\end{cases}
\end{align}
$$
Eliminating $\,i\,$ between the two equations, for example by taking $\,i=\alpha-\omega\,$ from the first equation and substituting into the second one:
$$
\alpha^2 - 2\omega\alpha + \omega^2+1=0
$$
Using that $\,\omega^2=-\omega-1\,$:
$$
\alpha^2 - 2\alpha\omega - \omega=0
$$
Multiplying by $\,\omega\,$ and using that $\,\omega^2=-\omega-1\,$:
$$
\require{cancel}
\begin{align}
\begin{cases}
\alpha^2 - \omega (2\alpha + 1) &= 0
\\ \omega\,\alpha^2 + (\omega+1)(2\alpha+1) &= 0
\end{cases}
\;\;\iff\;\; \begin{cases}
- \omega\,(2\alpha+1) + \alpha^2 &= 0
\\ \omega\,(\alpha^2+2\alpha+1) + 2\alpha+1&=0
\end{cases}
\end{align}
$$
Eliminating $\,\omega\,$ between the two equations:
$$
(\alpha^2+2\alpha+1)\alpha^2+(2\alpha+1)^2 = 0
 \;\;\;\;\iff\;\;\;\; \alpha^4 + 2 \alpha^3 + 5 \alpha^2 + 4 \alpha + 1 = 0
$$
With $\,P(x)=x^2+1\,$ and $\,Q(x)=x^2+x+1\,$, the above procedure is equivalent to calculating the polynomial resultant of $\,P(x)\,$ and $\,Q(\alpha-x)\,$, as verified in WA. A similar procedure can be used for higher degrees to produce a polynomial having as roots the sums of roots of $\,P\,$ and $\,Q\,$ with coefficients in the same ring as those of $P,Q$, in this case $\mathbb Z\,$.
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