Differential principal factors and Pólya property of pure metacyclic fields

Mayer, Daniel C.

doi:10.1142/s1793042119501094

Cited by 7 publications

(29 citation statements)

References 26 publications

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“…Precisely four of them can actually be given by parametrized infinite sequences in a deterministic way aside from the intrinsic probabilistic nature of the occurrence of primes in residue classes and of composite integers with assigned shape of prime decomposition. This was proved in [15,Thm. (1) f = q with q ∈ P, q ≡ ±7 (mod 25) gives rise to a singulet, m = 1, with DPF type ϑ,…”

Section: 3mentioning

confidence: 73%

“…Let f be the class field theoretic conductor of the relatively quintic Kummer extension N/K over the cyclotomic field K = Q(ζ). It is also called the conductor of the pure quintic field L. The multiplicity m = m(f ) of the conductor f indicates the number of non-isomorphic pure metacyclic fields N sharing the common conductor f , or also, according to [15,Prop. 2.1], the number of normalized fifth power free radicands D > 1 whose fifth roots generate non-isomorphic pure quintic fields L sharing the common conductor f .…”

Section: Collection Of Multiplicity Formulasmentioning

confidence: 99%

“…The normalized radicand D = q e1 1 · · · q es s of a pure metacyclic field N of degree 20 is minimal among the powers D n , 1 ≤ n ≤ 4, with corresponding exponents e j reduced modulo 5. The normalization of the radicands D provides a warranty that all fields are pairwise non-isomorphic [15,Prop. 2.1].…”

Section: 3mentioning

confidence: 99%

“…Principal factors, P, are listed when their constitution is not a consequence of the other information. According to [15,Thm. 7.2., item (1)] it suffices to give the rational integer norm of absolute principal factors.…”

Section: 3mentioning

confidence: 99%

“…The lack of a prime divisor ℓ ≡ ±1 (mod 5) together with the existence of a prime divisor q ≡ ±7 (mod 25) and q = 5 of D is indicated by a symbol × for the component 1. In these cases, only the two DPF types γ and ε can occur [15,Thm. 8.1].…”

Section: 3mentioning

confidence: 99%

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Tables of Pure Quintic Fields

Mayer¹

2019

APM

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By making use of our generalization of Barrucand and Cohn's theory of principal factorizations in pure cubic fields Q( 3 √ D) and their Galois closures Q(ζ 3 , 3 √ D) with 3 possible types to pure quintic fields L = Q( 5 √ D) and their pure metacyclic normal fields N = Q(ζ 5 , 5 √ D) 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(ζ 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 5vector 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 : U 0 ). The statistical distribution of the 13 principal factorization types and their refined splitting into similarity classes with representative prototypes is discussed thoroughly.

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Section: 3mentioning

confidence: 73%

Section: Collection Of Multiplicity Formulasmentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

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Tables of Pure Quintic Fields

Mayer¹

2019

APM

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Algebraic number fields generated by an infinite family of monogenic trinomials

Mayer¹,

Soullami

2022

Bol. Soc. Mat. Mex.

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Classifying multiplets of totally real cubic fields

Mayer¹

2021

Electronic Journal of Mathematics

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The number of non-isomorphic cubic fields L sharing a common discriminant dL = d is called the multiplicity m = m(d) of d. For an assigned value of d, these fields are collected in a homogeneous multiplet M d = (L1, . . . , Lm). By entirely new techniques for the construction and classification, we determine the differential principal factorization types τ (Li) ∈ {α1, α2, α3, β1, β2, γ, δ1, δ2, ε} of the members Li of each multiplet M d of non-cyclic totally real cubic fields with discriminants d < 10 7 . This is a new kind of arithmetical invariants which provide succinct information about ambiguous principal ideals and capitulation in the normal closures N of non-Galois cubic fields L. The classification is arranged with respect to increasing 3-class rank of the quadratic subfields K of the S3-fields N , and to ascending number of prime divisors of the conductor f of N/K. The Scholz conjecture concerning the distinguished index of subfield units (UN : U0) = 1 for ramified extensions N/K with conductor f > 1 is also verified by a tremendous minimal discriminant d = 18 251 060 outside of the systematic range d < 10 7 .

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Differential principal factors and Pólya property of pure metacyclic fields

Abstract: Barrucand and Cohn's theory of principal factorizations in pure cubic fields Q( 3 √ D) and their Galois closures Q(ζ 3 , 3 √ D) with 3 types is generalized to pure quintic fields L = Q( 5 √ D) and pure metacyclic fields N = Q(ζ 5 , 5

Cited by 7 publications

References 26 publications

Tables of Pure Quintic Fields

Tables of Pure Quintic Fields

Algebraic number fields generated by an infinite family of monogenic trinomials

Classifying multiplets of totally real cubic fields

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