Three stage towers of 5-class fields

Abstract

With K=Q((3812377)^(1/2)) we give the first example of an algebraic number field possessing a 5-class tower of exact length L(5,K)=3. The rigorous proof is conducted by means of the p-group generation algorithm, showing the existence of a unique finite metabelian 5-group G with abelianization [5,5] having the kernels (M(1),G^5) and targets ([25,5,5,5],[5,5]^5) of Artin transfers T(i):G-->M(i)/M(i)' to its six maximal subgroups M(i), prescribed by arithmetical invariants of K. Thus, G must be the second 5-class group G(5,2,K) of the real quadratic field K but cannot be its 5-class tower group G(5,K), since the relation rank d(2,G)=4 is too big. We provide evidence of exactly five non-isomorphic extensions H of G having the required relation rank d(2,H)=3 and derived length dl(H)=3 whose metabelianization H/H'' is isomorphic to G. Consequently, G(5,K) must be one of the five non-metabelian groups H.