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There is quite a lot of work on computing Galois groups of splitting fields of polynomials over $\Bbb Q$. Magma is quite good at it.

In this answer to Decomposition groups for the Galois module $\mu_8$ KConrad computes the decomposition groups over a number field in a special case using an ad hoc method. See also this answer to Biquadratic extension of global function fields with cyclic decomposition groups in which Will Sawin computes the decomposition groups over ${\Bbb F}_q(t)$ using an ad hoc method.

Question. Are there any computer programs computing the decomposition groups?

Motivation: I want to use computer to compute the Galois cohomology group $H^1(K,T)$ for an algebraic torus $T$ over a global field $K$. This more or less can be reduced to computation of $Ш^1(K,G)$ and $H^1(K_v,G)$, where $v$ runs over the places of $K$. To compute this guys, one needs decomposition groups (defined up to conjugation) as subgroups of ${\rm Aut}\,X_*(T)$, where $X_*(T)$ is the cocharacter group of $T$ (over a separable closure of $K$).

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    $\begingroup$ If I understand well, you want the decomposition group inside a global Galois group, not just a method to determine Galois groups of local fields. sage has K.galois_group().decomposition_group(p) for a number field K and a prime ideal p. $\endgroup$
    – Chris Wuthrich
    Commented May 30 at 22:30
  • $\begingroup$ @ChrisWuthrich: Yes, you ubderstand my question correctly. $\endgroup$
    – Mikhail Borovoi
    Commented May 31 at 6:59
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    $\begingroup$ Magma has DecompositionGroup(p) as found here. When you have calculated the Galois group it is easy to implement the action on prime ideals and finding what you are looking for. $\endgroup$
    – Chris Wuthrich
    Commented May 31 at 8:04
  • $\begingroup$ Thank you, @ChrisWuthrich! $\endgroup$
    – Mikhail Borovoi
    Commented May 31 at 8:18

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