论文标题
Schur \（σ\）的极端根路径 - 组和第一个\（3 \） - 具有四个阶段的阶级塔塔
Extremal root paths of Schur \(σ\)-groups and first \(3\)-class field towers with four stages
论文作者
论文摘要
有限的Schur Sigma-groups G的极端特性是用其在其阿贝利安化g/g'后代树中的根源的路径来描述的。通过3级野外塔的Galois组G = GAL（F（3，Infty，K）/K）的所有已知示例的所有已知示例的3级野外塔k = f（3,0，k）<f（3,1，k）<f（3,2,2，k）<= ...等级第二。这样的Galois组必须是Schur Sigma-groups，并且通过显示具有派生长度DL（g）<= 3的合适的Schur Sigma-groups G的不存在的不存在的塔的存在是合理的。通过反例，强调的是，具有相同类型的3级组类型的实际二次数字字段揭示了完全不同的行为，通常没有极端路径。
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.
