TITLE:
Tables of Pure Quintic Fields
AUTHORS:
Daniel C. Mayer
KEYWORDS:
Pure Quintic Fields, Pure Metacyclic Fields, Units, Galois Cohomology, Differential Principal Factorization Types, Similarity Classes, Prototypes, Class Group Structure
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.9 No.4,
April
29,
2019
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≤D3. Our classification is based on the Galois cohomology of the unit group UN, 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 (Uk:NN/K(UN)) by the number #(PN/K/PK) of primitive ambiguous principal
ideals, which can be interpreted as principal factors of the different DN/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 (UN:U0).The statistical distribution of the 13 principal factorization types
and their refined splitting into similarity classes with representative
prototypes is discussed thoroughly.