# Looking for some numerical examples of decomposition groups and ramification groups

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 find some worked out examples of these definitions. Where can I find them ?

• Harvey Cohn's Invitation to algebraic number fields contains lots of examples. – franz lemmermeyer Nov 8 '16 at 9:36
• You can look at "Hilbert sequences", for instance Computation of Hilbert sequence for composite quadratic extensions using different type of primes in $\Bbb Q$, by M. Haghighi and J. Miller. – Watson Jan 18 '17 at 15:33
• The examples in Harvey Cohn's book are for instance on page 202. – Watson Jan 18 '17 at 15:36