This last, and very short, chapter has one central result and some of its noteworthy implications. This result is due to Golod and Shafarevitch, published recently in a remarkable paper. Their theorem, rather easy to prove, provides a general method and technique for considering a large assortment of problems. As an immediate consequence of the main theorem one can construct a nil but not locally nilpotent algebra thereby giving a negative answer to the Kurosh Problem; one can construct a torsion group which is not locally finite thereby settling in the negative the general Burnside Problem. In addition the theorem gives rise to important results and examples in the theory of p-groups and in class field theory. However it seems likely that these successful uses of the method are merely a beginning, that a host of results await the application of the technique. We shall develop the results to such a point that we can construct the above-mentioned counterexamples to the Kurosh and Burnside Problems. We advise the reader to go to his original paper to see how the result is used in other algebraic areas.

Let F be any field and let T= F[x1, …, xd] be the polynomial ring over F in the d noncommuting variables x1, …, xd.