The inverse Galois problem with minimal ramification
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Various authors have studied variations of the inverse Galois problem over Q, which, in its classical form, asks for a finite group G if there exists a number field K such that its Galois group over Q is G. Boston and Markin conjecture that every group G should appear as the Galois group of a number field which ramifies at exactly d primes (counting the infinite prime), where d denotes the minimal number of generators of the abelianisation of G. They prove the validity of their conjecture for all groups up to order 32. We provide examples of this conjecture for groups with cyclic abelianisation and we illustrate how to use these to construct examples of order larger than 32. Harbater, Hoelscher and Pollak give a description of the Galois groups that can appear for number fields where a single predetermined prime is ramified. They respectively focus on the primes 2, 3 and 5 and we give a similar description of the possible Galois groups that can appear for the primes 5<p<23. Furthermore, Pollak proved for primes p<37 that, if G is a Galois group of a number field where only p ramifies and |G|<660, then G must be solvable. We strengthen this result by implementing one of the approaches of Pollak in GAP and obtaining an improved range of p<101 and a larger set of options for |G|.