# Is there any way to find decomposition group 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 compute the decomposition group and ramification groups of the cyclotomic field $\mathbb{Q}(\zeta)$ over $\mathbb{Q}$ where $\zeta$ is a root of unity.

How to do this ? Any idea ?

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Searching the web for ["decomposition group" galois extension sagemath] gives hints.

Here is how to compute the decomposition group and ramification group.

Setup.

sage: K = QQ
sage: L = CyclotomicField(4)
sage: G = L.galois_group()
sage: OK = K.ring_of_integers()
sage: OL = L.ring_of_integers()
sage: P = OK.ideal(5)
sage: Q = L.primes_above(5)[0]

Check setup.

sage: K
Rational Field
sage: L
Cyclotomic Field of order 4 and degree 2
sage: G
Galois group of Cyclotomic Field of order 4 and degree 2
sage: P
Principal ideal (5) of Integer Ring
sage: Q
Fractional ideal (-zeta4 - 2)

Decomposition group.

sage: Q.decomposition_group()
Subgroup [()] of Galois group of Cyclotomic Field of order 4 and degree 2
sage: Q.ramification_group(2)
Subgroup [()] of Galois group of Cyclotomic Field of order 4 and degree 2

Please suggest any edits to make the examples more interesting.

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