Tables of Pure Quintic Fields

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<10³. 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(ξ₅), 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 F₅-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_N:U₀)