Problem 15.3.4 in Artin's Algebra reads:
Let $\zeta_n = e^{2\pi i/n}$. Determine the irreducible polynomial over $\mathbb{Q}$ and over $\mathbb{Q}(\zeta_3)$ of:
(a) $\zeta_4$, (b) $\zeta_6$, (c) $\zeta_8$, (d) $\zeta_9$, (e) $\zeta_{10}$, (f) $\zeta_{12}$
Let's just consider the problem of finding the irreducible (minimal) polynomial over $\mathbb{Q}$ for now. It is well-known that the cyclotomic polynomials are irreducible, but at this point in the text this fact is only stated for prime powers, and the general proof is nontrivial. Also, Galois theory hasn't been covered yet.
It's obvious that $\Phi_4(x) = x^2 + 1$ and $\Phi_6(x) = x^2 - x + 1$ are indeed irreducible. The irreducibility of $\Phi_8(x) = x^4 + 1$, $\Phi_{9}(x) = x^6 + x^3 + 1$, and $\Phi_{10}(x) = x^4 - x^3 + x^2 - x + 1$ are less obvious but can be shown by shifting and applying the Eisenstein criterion.
For $n = 12$ the cyclotomic polynomial is $x^4 - x^2 + 1$. In this case we could simply compute the roots and show that none of the linear or quadratic factors lie in $\mathbb{Q}[x]$, but already this is a bit tedious, and I am wondering if there is some elementary approach I'm missing.
When we get to the question of the minimal polynomials over $\mathbb{Q}(\zeta_3)$, it is helpful to know the irreducibility of cyclotomic polynomials of even higher order. For example, the technique given in this answer can be applied to show that $\Phi_{10}$ is irreducible over $\mathbb{Q}(\zeta_3)$ but only if we know that $\zeta_{30}$ indeed has degree 8 over $\mathbb{Q}$.
What do you think are the solutions that Artin intended to be found by a student who has only read up to this point in the text?