Elements and structure of $\text{Gal}(\mathbb Q(\zeta_{21}):\mathbb Q)$

Question

Let $\zeta_{21}$ be a primitive 21st root of 1. Describe explicitly the elements of the Galois group $\text{Gal}(\mathbb Q(\zeta_{21}):\mathbb Q)$ and determine the structure of this group.

From finding the minimal polynomial I know that the order of the Galois group must be 12. So $\text{Gal}(\mathbb Q(\zeta_{21}):\mathbb Q)≅(\mathbb Z/21\mathbb Z)^{\times}$.

I have seen online that $(\mathbb Z/21\mathbb Z)^{\times}$ has the structure $C_2 \times C_6$, how has this been found?

How can I describe the elements of this Galois group?

Answer 1

I suppose you mean $U(21)$. This has $\phi(21)=12$ elements, and is not cyclic:

For what $n$ is $U_n$ cyclic?

From $\mathbb{Z}/21\cong \mathbb{Z}/3\times \mathbb{Z}/7$ we see that $U(21)\cong U(3)\times U(7)\cong C_2\times C_6$. Here we have used that $U(p)$ is cyclic of order $p-1$, see the above link.