Constructing Maximal Unramified Extensions and Murphy's Law for Galois Deformation Rings

The structure of the Galois group of the maximal unramified extension of a number field has been an object of interest for more than a century now. This thesis is partially motivated by the question of which finite groups can appear as quotients of this Galois group. We prove, under a technical assumption, that any semi-direct product of a p-group with a group of order prime to p can appear as the Galois group of a tower of extensions M/L/K with the property that M is the maximal unramified p-extension of L and the Galois group of M/L is isomorphic to the p-group. A consequence of this result is that any local ring admitting a surjection onto the 5-adic or the 7-adic integers with finite kernel can occur as a universal unramified deformation ring.