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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.