ABSTRACT
Our goal will be to prove the following theorem which is due to Scholz and Reichardt [Re]:
Theorem 2.1.1 Every l-group, l = 2, can be realized as a Galois group over Q. (Equivalently, every finite nilpotent group of odd order is a Galois group over Q.)
An l-group can be built up from a series of central extensions by groups of order l. The natural approach to the problem of realizing an l-group G as a Galois group over Q is to construct a tower of extensions of degree l which ultimately give the desired G-extension. When carried out naively, this approach does not work, because the embedding problem cannot always be solved. The idea of Scholz and Reichardt is to introduce more stringent conditions on the extensions which are made at each stage, ensuring that the embedding problem has a positive answer.
