We describe an algorithmic approach to prove or disprove several recent conjectures for epsilon constants of Galois extensions of $p$-adic fields and number fields. For this approach, we must develop various algorithms for computations in Galois extensions of $p$-adic fields which are of independent interest. Our algorithms for $p$-adic fields are based on existing algorithms for number fields and are exact in the sense that we do not need to consider approximations to $p$-adic numbers.