Find a degree 3 polynomial that has a galois group of the extension field which is isomorfic to the ciclic group of order 3.

Question

I'm looking for a polynomial $f \in \mathbb{Q}$ of degree $3$ that is irreducible over $\mathbb{Q}$, such that the polynomial's splitting field $L/\mathbb{Q}$ has a Galois group $G\text{al}(L/\mathbb{Q}) = G$ with $G$ isomorfic to the cyclic group of order 3.

The question actually asks me to give this polinomial in a seventh root of unity $\zeta$ with $\zeta^7 = 1$. I guess i should be able to write it in $\mathbb{Q}(\zeta)$.

Could anyone help me find this polynomial, or give me a hint?

Thanks!

Answer 1

Note that $\mathbb{Q}(\zeta_7)$ is definitely not quite the extension you want, since that gives you the whole splitting field of 7-th cyclotomic polynomial, which has degree $\phi(7) = 6$. However you can find the extension you're looking for as an intermediate extension of $\mathbb{Q}(\zeta_7)/\mathbb{Q}$. A nice way to approach this is to note that in order to get an extension of degree three, all roots must be real, or in other words the extension must be fixed by complex conjugation.