Algorithms to compute Galois groups of irreducible polynomials over the
rational field have been available in some way for some time. These algorithms
have been extended to polynomials of larger degrees and also polynomials over
other coefficient rings.
Currently the widest ranging algorithm
is that of Fieker and Kl\"uners which has no degree restriction on input
polynomials and has been adapted for use with reducible as well as irreducible
polynomials over algebraic
number fields, rational function fields of all characteristics and global
algebraic function fields.
In this talk I will briefly summarise this algorithm and discuss how we
can do further computations with Galois groups using the information we have
from the initial computation.