The Inverse Galois Problem for Nilpotent Groups of Odd Order

Consider any nilpotent group G of finite odd order. We ask if we can always find a galois extension K of Q such that Gal(K/Q) ∼= G. This is the famous Inverse Galois Problem applied to nilpotent groups of finite odd order. By solving the Group Extension Problem and the Embedding Problem, two problems that are related to the Inverse Galois Problem, we show that such a K always exists. A major result of Shafarevich tells us that such a K exists for all solvable groups G, but the proof is far too difficult to be presented here. Nevertheless, we present Shafarevich’s results and a sketch of the main idea. We then show that for the group Sn, one can use elementary techniques in Galois Theory to solve the Embedding Problem, constructing a solution to the Inverse Galois Problem for this group as well.