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Tables of Pure Quintic Fields

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Document pages: 57 pages

Abstract: By making use of our generalization of Barrucand andCohn’s theory of principal factorizations in pure cubic fields and their Galois closures with 3 possible types to purequintic fields and their pure metacyclic normalfields with 13 possible types, we compilean extensive database with arithmetical invariants of the 900 pairwisenon-isomorphic fields N havingnormalized radicands in the range 2≤D3. Our classification is based on the Galois cohomology of the unit group UN, viewed as a module overthe 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 ofthe unit norm index (Uk:NN K(UN)) by the number #(PN K PK) of primitive ambiguous principalideals, 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 ofnorm kernels and central orthogonal idempotents. A connection with integralrepresentation theory is established via class number relations by Parry andWalter involving the index of subfield units (UN:U0). The statistical distribution of the 13 principal factorization typesand their refined splitting into similarity classes with representativeprototypes is discussed thoroughly.

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