Tables of Pure Quintic Fields - Advances in Pure Mathematics

APM > Vol.9 No.4, April 2019

Tables of Pure Quintic Fields ()

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DOI: 10.4236/apm.2019.94017 499 Downloads 1,029 Views Citations

Author(s)

Daniel C. Mayer

Affiliation(s)

Naglergasse 53, Graz, Austria.

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<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)_. The statistical distribution of the 13 principal factorization types and their refined splitting into similarity classes with representative prototypes is discussed thoroughly.

KEYWORDS

Pure Quintic Fields, Pure Metacyclic Fields, Units, Galois Cohomology, Differential Principal Factorization Types, Similarity Classes, Prototypes, Class Group Structure

Share and Cite:

Mayer, D. (2019) Tables of Pure Quintic Fields. Advances in Pure Mathematics, 9, 347-403. doi: 10.4236/apm.2019.94017.

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