2016/04/23 by Daniel C. Mayer, Mayer, Daniel C.
Computer Science · Mathematics · #11R11 #11R20 #11R29 #11R37 #20-04 #20D15 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1604.06930
openalex publication_date 2016/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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),G5) 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.