Let $\alpha =e^{2i\pi/7}$. Prove that $\Bbb Z[\alpha ]$ is principal.
I try to prove that the class group of $\Bbb K:=\Bbb Q[\alpha ]$ is trivial which I think is sufficient condition to prove that $\mathcal O_K$ is principal.
The Minkowski bound computed on Sage is $M_k=4.12$ for $n:=[K:\Bbb Q]=6, r_1=0, r_2=3$.
We need to check the prime ideals of $\mathcal O_K$ lying above $(2)$ and $(3)$.
Let $f(x)=1+x+x^2+x^3+x^4+x^5+x^6$ be the minimal polynomial of $\alpha $ over $\Bbb Q$.
since $x\equiv x^2\equiv x^3\equiv x^4\equiv x^6 (mod\ 2)$, reducing $f$ mod 2 we get:
$f_2(x)\equiv 1+x^3+x^5=1+x(x^2+x^4)\equiv 1\ (mod\ 2)$
Hence $2\mathcal O_K=(2,1)=(1)$
I am not sure how to interpret this result. I think that something is wrong since in sage I get a different result:
$ (Fractional\ ideal (\alpha ^3 + \alpha + 1)) * (Fractional\ ideal (-\alpha ^5 - \alpha ^3 - \alpha ^2 - \alpha ))$.
Moving on with factoring $(3)$
$x\equiv x^3\equiv x^5 (mod\ 3) \Rightarrow f_3(x)\equiv 1+x^2+x^6=1-2x^2-2x^6$ which is irreducible by Eisenstein at the prime $2$.
Hence in $3\mathcal O_K=(3,1+\alpha ^2+\alpha ^6)$ is prime
So Since $(2)$ and $(3)$ are also prime in $\mathcal O_K$ so the ideal classes $[2]$ and $[3]$ generate the class group. But since the representatives $(2)$ and $(3)$ are are principal, they both belong to the trivial ideal class, Hence $\mathcal O_K$ is principal.
Is this correct? Thank your for any help