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This is a common exercise:

Sketch the lattice of subfields of $F = \mathbb{Q} ( \mathbb{e^{\frac{2 \pi i}{p}}})$ be a cyclotomic extension over $\mathbb{Q}$ (where $p$ is an odd prime).

It got me wondering, what's the easiest way of writing/listing down the elements of the Galois group $Aut_{\mathbb{Q}} F$?

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Let $\zeta = \mathbb{e^{\frac{2 \pi i}{p}}}$. Let $n$ be any integer not divisible by $p$. There exists a unique $\sigma \in Aut_{\mathbb{Q}} F$ such that $\sigma(\zeta) = \zeta^n$. Let us denote this $\sigma$ by $\sigma(n)$. Every element of $Aut_{\mathbb{Q}} F$ can be written in this form. If $n \equiv m$ (mod $p$), $\sigma(n) = \sigma(m)$. Hence we get a map $(\mathbb{Z}/p\mathbb{Z})^* \rightarrow Aut_{\mathbb{Q}} F$, where $(\mathbb{Z}/p\mathbb{Z})^*$ is the multiplicative group of the field $\mathbb{Z}/p\mathbb{Z}$. This map is a group isomorphism. It is well known that $(\mathbb{Z}/p\mathbb{Z})^*$ is a cyclic group. Let $r$ be a primitive root mod $p$, i.e. $r$ (mod $p$) be a generator of the group $(\mathbb{Z}/p\mathbb{Z})^*$. Then $\sigma(r)$ is a generator of $Aut_{\mathbb{Q}} F$.

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