Irreducible polynomial of ζ_6, ζ_8, ζ_9 over the field Q(ζ_3).

[kalish1]

How can I get started on finding the irreducible polynomial of $\zeta_6, \zeta_8, \zeta_9$ over the field over $\mathbb{Q}(\zeta_3)?$ Should I construct field extensions and then use the degrees of the extensions?

This question has been crossposted here: abstract algebra - Finding the irreducible polynomial of $\zeta_6, \zeta_8, \zeta_9$ over the field $\mathbb{Q}(\zeta_3).$ - Mathematics Stack Exchange

 

[jvicens]

Hello,

Thank you for your question. Finding the irreducible polynomial of $\zeta_6, \zeta_8, \zeta_9$ over the field $\mathbb{Q}(\zeta_3)$ can be approached in a few different ways. One possible method is indeed to construct field extensions and use the degrees of the extensions to determine the irreducible polynomial. However, there are also other methods that can be used.

One approach is to use the fact that $\zeta_n$ is a primitive $n$th root of unity, meaning that it is a complex number that, when raised to the $n$th power, gives the result of 1. Using this property, we can write $\zeta_6 = e^{2\pi i/6} = \cos(\pi/3) + i\sin(\pi/3) = 1/2 + i\sqrt{3}/2$. Similarly, $\zeta_8 = e^{2\pi i/8} = \cos(\pi/4) + i\sin(\pi/4) = \sqrt{2}/2 + i\sqrt{2}/2$, and $\zeta_9 = e^{2\pi i/9} = \cos(2\pi/9) + i\sin(2\pi/9)$.

Using these expressions, we can then construct a polynomial of degree 6 that has $\zeta_6$ as a root, a polynomial of degree 8 that has $\zeta_8$ as a root, and a polynomial of degree 9 that has $\zeta_9$ as a root. These polynomials will necessarily be irreducible, as they will have the minimal degree needed to have these roots.

Another approach is to use the fact that the minimal polynomial of $\zeta_n$ over $\mathbb{Q}$ is the cyclotomic polynomial $\Phi_n(x) = \prod_{d|n} (x^d - 1)^{\mu(n/d)}$, where $\mu$ is the Möbius function. This means that we can find the irreducible polynomial of $\zeta_n$ over $\mathbb{Q}$ by finding the factors of $\Phi_n(x)$ that are irreducible over $\mathbb{Q}$. For example, the cyclotomic polynomial for $n=6$ is $\Phi_