Finding the degree of $\Bbb Q(\zeta_9)$

Question

I know that the degree is $6$. But now I need to prove it without using Galois theory or the theorem about the function $\phi(n)$. May I ask a proof with an explicit minimal polynomial of $\zeta_9$ over $\Bbb Q$ with justification? Thanks!

Answer 1

The minimal polynomial is $$x^6 + x^3 +1.$$ That the polynomial is irreducible over $\mathbb{Q}$, follows from the fact that it is irreducible mod 2.

Following Kaj Hansens's comment: To show that the polynomial is irreducible, you have to check that it has no roots (mod 2) and it cannot be factored by an irreducible polynomial of degree 2 or 3 (mod 2). There are not so many of those. A complete list is the following: $$x^2+x+1, \quad x^3+x^2 +1,\quad x^3+x +1.$$