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Let $L/K$ be a Galois extension of number fields with Galois group $G$. Let $O_K$ and $O_L$ be the ring of algebraic integers of $K$ and $L$ respectively. Let $P\subseteq O_K$ be a prime. Let $Q\subseteq O_L$ be a prime lying over $P$.

The decomposition group is defined as $$D(Q|P)=\lbrace \sigma\in G\text{ }|\text{ }\sigma(Q)=Q\rbrace$$

The $n$-th ramification group is defined as $$E_n(Q|P)=\lbrace \sigma\in G:\sigma(a)\equiv a\text{ mod } Q^{n+1}\text{ for all } a\in O_L\rbrace$$

I want to compute the decomposition group and ramification groups of the cyclotomic field $\mathbb{Q}(\zeta)$ over $\mathbb{Q}$ where $\zeta$ is a primitive $p$-th root of unity ($p$ is a prime).

In general, how to calculate it for an arbitrary cyclotomic field ?

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    $\begingroup$ Do you want to compute the decomposition group and ramification groups of a prime ideal of the cyclotomic field $\Bbb Q(\zeta_p)$? For instance a prime ideal over the integer prime $p$? You question isn't clear otherwise (in my opinion). $\endgroup$ – Watson Nov 7 '16 at 17:05
  • $\begingroup$ Yes, that is exactly what I want. $\endgroup$ – learning_math Nov 7 '16 at 17:16
  • $\begingroup$ Since $p$ is totally ramified in $\Bbb Q(\zeta_p)$, it follows that $D(P|p) = \mathrm{Gal}(K/\Bbb Q) =: G$ where $P$ is a prime ideal of $\mathcal O_K$ above $p$ and $K=\Bbb Q(\zeta_p)$, since the order of $D(P)$ is $e_{P \mid p} \cdot f_{P \mid p}$. $\endgroup$ – Watson Nov 7 '16 at 17:18
  • $\begingroup$ What about the ramification groups ? $\endgroup$ – learning_math Nov 7 '16 at 17:20
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    $\begingroup$ See Sublemma 1 in An Elementary Proof of the Kronecker-Weber Theorem by M. J. Greenberg : if $K/\Bbb Q$ is abelian of degree $p$ (odd prime number), such that $p$ is the only ramified prime, then the second ramification group $E_2(P/p)$ is trivial (for any $P$ over $p$). $\endgroup$ – Watson Jan 4 '17 at 14:12

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