Teorema de Golod-Shafarevich

Abstract

This thesis is about the celebrated theorem of Golod and Shafarevich. It can be viewed as a theorem on homological algebra for noncommutative local rings. However, the main motivation in nding and proving it came from number theory: if K is an algebraic number eld (i.e., a nite extension of Q), and we iterate the construction of the Hilbert class eld (the maximal abelian unrami ed extension of K), we get the class eld tower of K: The theorem shows that in general such towers can be in nite. For instance, when the discriminant of K=Q has at least 6 prime factors, and K is imaginary quadratic, then such tower is always in nite. For more general extensions it is still true that those towers can be in nite, provided that that discriminant has a large enough number of prime factors. Another related area where the theorem is relevant is group theory. If G is a nite p-group with d being its minimal number of generators, and r relations, the theorem asserts that r > d2=4: The connection with group cohomology comes from the fact that d = dim H1(G;Z=pZ) and r = dim H2(G;Z=pZ): The theorem has an interesting application to the generalised Burnside conjecture. For each prime p there is an in nite group generated by 3 elements, in which every element is of nite order, namely a power of p: The main tools used came from homological algebra, especially group homology and cohomology.