Extremal root paths of Schur \(σ\)-groups and first \(3\)-class field towers with four stages

Abstract: An extremal property of finite Schur sigma-groups G is described in terms of their path to the root in the descendant tree of their abelianization G/G'. The phenomenon is illustrated and verified by all known examples of Galois groups G=Gal(F(3,infty,K)/K) of 3-class field towers K=F(3,0,K)<F(3,1,K)<F(3,2,K)<=...<=F(3,infty,K) of imaginary quadratic number fields K=Q(sqrt{d}), d<0, with elementary 3-class group Cl(3,K) of rank two. Such Galois groups must be Schur sigma-groups and the existence of towers with at least four stages is justified by showing the non-existence of suitable Schur sigma-groups G with derived length dl(G)<=3. By means of counter-examples, it is emphasized that real quadratic number fields with the same type of 3-class group reveal a totally different behavior, usually without extremal path.