Tables of Pure Quintic Fields

Abstract

By making use of our generalization of Barrucand and Cohn’s theory of principal factorizations in pure cubic fields and their Galois closures with 3 possible types to pure quintic fields and their pure metacyclic normal fields with 13 possible types, we compile an extensive database with arithmetical invariants of the 900 pairwise non-isomorphic fields N having normalized radicands in the range 2≤D<10 3 . Our classification is based on the Galois cohomology of the unit group U N , viewed as a module over the automorphism group Gal(N/K) of N over the cyclotomic field K=Q(ξ 5 ) , by employing theorems of Hasse and Iwasawa on the Herbrand quotient of the unit norm index (U k :N N/K (U N )) by the number #(P N/K /P K ) of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different D N/K . The precise structure of the F 5 -vector space of differential principal factors is expressed in terms of norm kernels and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units (U<SUB>N</SUB>:U<SUB>0</SUB>) . The statistical distribution of the 13 principal factorization types and their refined splitting into similarity classes with representative prototypes is discussed thoroughly.