Finding irreducible polynomials in a field.

Question

I came across the following problem in Dummit and Foote which states:

Find an irreducible polynomial for $e^{2\pi i/9}$ and $e^{2\pi i/10}$ over $\mathbb Q(e^{2\pi i/3})$ and $\mathbb Q$.

So we know $\mathbb Q(e^{2\pi i/3})= \{a+be^{2\pi i/3}: a,b \in \mathbb Q\}$. So, in this case our variable "$x$" will be "$e^{2\pi i/3}$". Hence I am trying $e^{2\pi i/3}=e^{2\pi i/9}$ and $e^{2\pi i/3}=e^{2\pi i/10}$ However I am stuck beyond this point.Is there any easy way to find the polynomials other than guess and check? Also for $\mathbb Q$ my strategy will be guess and check but I am not sure if that is the right one.

Answer 1

Note that $(e^{2\pi i/9})^3=e^{2\pi i/3}$, so $f(X)=X^3-e^{2\pi i/3}$ is some polynomial in $\mathbb Q(e^{2\pi i/3})[X]$ that has $e^{2\pi i/9}$ as root. It is irreducible because otherwise it would have a linear factor, which it doesn't (because we can see all three roots).

Similarly, $(e^{2\pi i/10})^5=e^{\pi i}=-1$, so $f(X)=X^5+1$ is some polynomial with the required root - but is it irreducible?