3-rank of ambiguous class groups of cubic Kummer extensions

Aouissi, Siham; Mayer, Daniel C.; Ismaili, Moulay Chrif; Talbi, Mohamed; Azizi, Abdelmalek

doi:10.1007/s10998-020-00326-1

Cited by 9 publications

(31 citation statements)

References 19 publications

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“…then by Lemma 2.8 we have either 3 is not divide |C L,3 | or u = 3. Assume that 3 is not divide |C L,3 |, then according to[16,§1, Thm, p. 8] the prime p is congruent to −…”

mentioning

confidence: 99%

On a conjecture of Lemmermeyer

Aouissi

Talbi

Ismaili

et al. 2020

Int. J. Number Theory

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“…then by Lemma 2.8 we have either 3 is not divide |C L,3 | or u = 3. Assume that 3 is not divide |C L,3 |, then according to[16,§1, Thm, p. 8] the prime p is congruent to −…”

mentioning

confidence: 99%

On a conjecture of Lemmermeyer

Aouissi

Talbi

Ismaili

et al. 2020

Int. J. Number Theory

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“…Table 4. Differential principal factorization types, T, of pure metacyclic fields N T U η ζ A I R α 1…”

Section: Collection Of Multiplicity Formulasmentioning

confidence: 99%

“…The quartet (1,2,4,5) indicates conditions which either enforce a reduction of possible DPF types or enable certain DPF types. 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.…”

Section: 3mentioning

confidence: 99%

“…DPF type ε splits into 3 similarity classes in the ground state (V L , V M , V N ) = (0, 0, 0), 16 similarity classes in the first excited state (V L , V M , V N ) = (1,2,4), and 3 similarity classes in the second excited state (V L , V M , V N ) = (2,4,8). Summing up the partial frequencies 139 + 61 + 8 of these states in Table 39 yields the high absolute frequency 208 of type ε in the range 2 ≤ D < 10 3 , as given in Table 26.…”

Section: Refinement Of Dpf Types By Similarity Classesmentioning

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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“…be the Hilbert 3 -class field of k . When the 3 -class group C k,3 of k is of type (9,3) , the extension k…”

Section: Introductionmentioning

confidence: 99%

3-Principalization over S3 -fields

AOUISSI,

TALBI,

MAYER

et al. 2021

Turk J Math

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Let p ≡ 1 (mod 9) be a prime number and ζ3 be a primitive cube root of unity. Then k = Q( 3 √ p, ζ3) is a pure metacyclic field with group Gal(k/Q) S3 . In the case that k possesses a 3 -class group C k,3 of type (9, 3) , the capitulation of 3 -ideal classes of k in its unramified cyclic cubic extensions is determined, and conclusions concerning the maximal unramified pro-3 -extension k, that is the 3 -class field tower of k , are drawn.

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3-rank of ambiguous class groups of cubic Kummer extensions

Cited by 9 publications

References 19 publications

On a conjecture of Lemmermeyer

On a conjecture of Lemmermeyer

Tables of Pure Quintic Fields

3-Principalization over S3 -fields

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