7. [15] Find a real, irreducible quartic polynomial with leading coefficient 1 whose roots are all twelfth roots of unity.
Solution: All twelfth roots of unity are roots of
$$
\begin{aligned}
x^{12}-1 & =\left(x^{6}-1\right)\left(x^{6}+1\right) \\
& =\left(x^{3}-1\right)\left(x^{3}+1\right)\left(x^{6}+1\right) \\
& =(x-1)\left(x^{2}+x+1\right)(x+1)\left(x^{2}-x+1\right)\left(x^{2}+1\right)\left(x^{4}-x^{2}+1\right),
\end{aligned}
$$
so the answer is $x^{4}-x^{2}+1$.