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How to prove that the Galois group of the cyclotomic field $\mathbb{Q}(\zeta_n)$ is isomorphic to $U(\mathbb{Z}/n\mathbb{Z})$? The isomorphism is given by $U(\mathbb{Z}/n\mathbb{Z}){\rightarrow} Gal(\mathbb{Q(\zeta_n)/\mathbb{Q}})$, such that $\overline{a}\rightarrow \sigma_a$, where $\sigma_a$ is the automorphism defined by $\sigma_a(\zeta_n)=\zeta_{n}^{a}$.

i know this for the part of homomorphism, that is it is trivial to note(with a spoiler hint from Dummit and Foote) $$(\sigma_a \sigma_b)(\zeta_n)=\sigma_a(\zeta_{n}^{b})=(\zeta_{n}^{b})^a=\zeta_{n}^{ab}$$ But how do i show bijection? I am having difficuty in here. Can you please post a complete proof? It will also help me gain the insights. Thanks!

  • For each $\sigma$ then $\sigma(\zeta_n) $ is primitive $n$-th root of unity so $\sigma(\zeta_n) = \zeta_n^a$ for some $a \in (\Bbb{Z/nZ})^\times$. That $Gal(\Bbb{Q}(\zeta_n)/\Bbb{Q})$ is the whole of $(\Bbb{Z/nZ})^\times$ follows from the irreducibility of the cyclotomic polynomial $\Phi_n(x)=\prod_{a \in (\Bbb{Z/nZ})^\times} (x-\zeta_n^a)$, there are many proofs, none of them is trivial. – reuns Apr 29 '19 at 14:51
  • Thanks but can you answer it fully? –  Apr 29 '19 at 14:53

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