arXiv subjects

Daniel C. Mayer

Publications and source records attributed to Daniel C. Mayer.

At least 19 recordsLinked to original sources

arXiv2026

3-class field towers with 2 or 3 stages

For quadratic fields $k=\mathbb{Q}(\sqrt{d})$ with discriminant $d$, $3$-class group $\mathrm{Cl}_3(k)\simeq (\mathbb{Z}/3\mathbb{Z})^2$, and four \textit{simple} $3$-principalization types $\varkappa(k)\in\lbrace (1122),(3122),(1231),(2231)\rbrace$, we establish necessary and sufficient conditions for the Galois group $S=\mathrm{Gal}(\mathrm{F}_3^\infty(k)/k)$ of the unramified Hilbert $3$-class field tower of $k$ to coincide with the Galois group $M=\mathrm{Gal}(\mathrm{F}_3^2(k)/k)$ of the maximal metabelian unramified $3$-extension of $k$. In the case of non-coincidence, we study the path between $M$ and $S$ in the descendant tree of the elementary bicyclic $3$-group $(\mathbb{Z}/3\mathbb{Z})^2$. For two \textit{complex} $3$-principalization types $\varkappa(k)\in\lbrace (2122),(4231)\rbrace$, we show that infinitely many non-metabelian possible Galois groups $S=\mathrm{Gal}(\mathrm{F}_3^\infty(k)/k)$ with presumably unbounded derived length $\mathrm{dl}(S)$ share a common metabelianization $M=S/S^{\prime\prime}$, whence only partial criteria can be stated. Minimal discriminants $d>0$ with assigned simple $3$-principalization type $\varkappa(k)$ and fixed length $\ell_3(k)\in\lbrace 2,3\rbrace$ of the $3$-class field tower are determined experimentally for nilpotency class $\mathrm{cl}(M)\in\lbrace 5,7,9,11\rbrace$ under assumption of the generalized Riemann hypothesis.

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arXiv2025

Coclass of the second 3-class group

By means of parametrized presentations of finite metabelian 3-groups, it is proved that the coclass cc(M) of the second 3-class group M=Gal(F_3^2(K)/K) of any algebraic number field K with elementary bicyclic 3-class group Cl_3(K)=(3,3) is determined unambiguously by the second largest order ord(Cl_3(E_2))=3^{cc(M)+1} among the four 3-class groups of the unramified cyclic cubic extensions E_i (i=1,..,4) of K. Minimal discriminants of quadratic and cubic fields K with assigned coclass cc(M) are computed from extensive databases of 3-class numbers ord(Cl_3(E_i)) as an application.

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arXiv2025

The Capitulation Problem in Certain Pure Cubic Fields

Let $\Gamma=\mathbb{Q}(\sqrt[3]{n})$ be a pure cubic field with normal closure $k=\mathbb{Q}(\sqrt[3]{n},\zeta)$, where $n>1$ denotes a cube free integer, and $\zeta$ is a primitive cube root of unity. Suppose $k$ possesses an elementary bicyclic $3$-class group $\mathrm{Cl}_3(k)$, and the conductor of $k/\mathbb{Q}(\zeta)$ has the shape $f\in\lbrace pq_1q_2,3pq,9pq\rbrace$ where $p\equiv 1\,(\mathrm{mod}\,9)$ and $q,q_1,q_2\equiv 2,5\,(\mathrm{mod}\,9)$ are primes. It is disproved that there are only two possible capitulation types $\varkappa(k)$, either type $\mathrm{a}.1$, $(0000)$, or type $\mathrm{a}.2$, $(1000)$. Evidence is provided, theoretically and experimentally, of two further types, $\mathrm{b}.10$, $(0320)$, and $\mathrm{d}.23$, $(1320)$.

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arXiv2024

Some results about entropy and divergence in number theory

We obtain inequalities involving the entropy of a positive integer and the divergence of two positive integers, respectively the entropy of an ideal and the divergence of two ideals in a ring of algebraic integers. Among the important results, we show that the minimal entropy arises for sharp localization, and the maximal entropy occurs for equidistribution. We also study other interesting estimates of entropy and divergence for numbers and for ideals. Finally, we determine the entropies of probability distributions on infinite trees of Schur {\sigma}-groups, which are realized by 3-class field tower groups of imaginary quadratic number fields.

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arXiv2024

Group theory of cyclic cubic number fields

Astonishing new discoveries with quartets and octets of cyclic cubic fields sharing a common conductor are presented. Four kinds of graphs describing cubic residue conditions among the prime divisors of the conductor enforce elementary bi- or tricyclic 3-class groups and either a metabelian 3-class field tower group of coclass at least two or a closed Andozhskii-Tsvetkov group of order 6561. In the latter situation, abelian type invariants of first and second order are required for the identification.

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arXiv2024

Group theoretic approach to cyclic cubic fields

Let (k1,k2,k3,k4) be a quartet of cyclic cubic number fields sharing a common conductor c=pqr divisible by exactly three prime(power)s p,q,r. For those components k of the quartet whose 3-class group Cl(3,k) = Z/3Z x Z/3Z is elementary bicyclic, the automorphism group M = Gal(F(3,2,k)/k) of the maximal metabelian unramified 3-extension of k is determined by conditions for cubic residue symbols between p,q,r and for ambiguous principal ideals in subfields of the common absolute 3-genus field k* of k1,k2,k3,k4. With the aid of the relation rank d2(M), it is decided whether M coincides with the Galois group G = Gal(F(3,infinity,k)/k) of the maximal unramified pro-3-extension of k.

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arXiv2023

Structure of relative genus fields of cubic Kummer extensions

Let $N=K(\sqrt[3]{D})$ be a cubic Kummer extension of the cyclotomic field $K=\mathbb{Q}(ζ_3)$, containing a primitive cube root of unity $ζ_3$, with cube free integer radicand $D>1$. Denote by $f$ the conductor of the abelian extension $N/K$, and by $N^{\ast}$ the relative genus field of $N/K$. The aim of the present work is to find out all positive integers $D$ and conductors $f$ such that the genus group $\operatorname{Gal}\left(N^{\ast}/N\right)\cong \mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$ is elementary bicyclic.

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arXiv2023

Cyclic cubic number fields with harmonically balanced capitulation

It is proved that c = 689347 = 31*37*601 is the smallest conductor of a cyclic cubic number field K whose maximal unramified pro-3-extension E = F(3,infinity,K) possesses an automorphism group G = Gal(E/K) of order 6561 with coinciding relation and generator rank d2(G) = d1(G) = 3 and harmonically balanced transfer kernels kappa(G) in S(13).

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arXiv2023

Theoretical and Experimental Approach to p-Class Field Towers of Cyclic Cubic Number Fields

Cyclic number fields of odd prime degree are constructed as ray class fields over the rational number field. They are collected in multiplets sharing a common conductor and discriminant. The algorithms are implemented in Magma and applied to all cyclic quintic and cyclic cubic fields with conductors below 100000. Our primary attention is devoted to the theory of cyclic cubic fields with two or three prime divisors of the conductor. These fields form doublets and quartets. Theoretical techniques comprise cubic residue conditions between the primes dividing the conductor, the structure of 3-class groups of all components of doublets and quartets, Galois cohomology of unit groups and ambiguous principal ideals, absolute genus fields and their bicyclic bicubic subfields, class number relations, transfer kernels and abelian quotient invariants of unramified cyclic cubic extensions and their impact on the class field tower, pattern recognition via Artin transfers on descendant trees of finite groups with order a power of 3, the Shafarevich Theorem on the relation rank of the 3-class field tower group, and the Galois action on the tower group and on its metabelianization. Rigorous proofs are given for the first occurrences of three-stage towers over cyclic cubic fields with elementary bicyclic or tricyclic or non-elementary bicyclic 3-class group. Experimentally, the second p-class groups and the length of the p-class tower are determined for all conductors below 100000 and for p=2,3,5, with the exception of the few intricate octets. An interesting application is able to identify and realize the closed groups by Andozhskii and Tsvetkov.

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arXiv2022

Schur sigma-Groups of Scholz-Taussky Type F

For finite metabelian 3-groups M with elementary bicyclic commutator quotient M/M' = C3*C3, coclass cc(M) in {4,6}, and transfer kernel type F, the smallest Schur sigma-groups S with second derived quotient S/S" = M are determined. Evidence is provided of arithmetical realizations of these groups by second 3-class groups M = Gal(F(3,2,K)/K), respectively 3-class field tower groups S = Gal(F(3,infty,K)/K), of imaginary quadratic number fields K=Q(sqrt{d}).

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arXiv2022

Algebraic number fields generated by an infinite family of monogenic trinomials

For an infinite family of monogenic trinomials $P(X) = X^3\pm 3rbX-b$ in $\mathbb{Z}\lbrack X\rbrack$, arithmetical invariants of the cubic number field $L = \mathbb{Q}(θ)$, generated by a zero $θ$ of $P(X)$, and of its Galois closure $N = L(\sqrt{d(L)})$ are determined. The conductor $f$ of the cyclic cubic relative extension $N/K$, where $K = \mathbb{Q}(\sqrt{d(L)})$ denotes the unique quadratic subfield of $N$, is proved to be of the form $3^eb$ with $e\in\lbrace 1,2\rbrace$, which admits statements concerning primitive ambiguous principal ideals, lattice minima, and independent units in $L$. The number $m$ of non-isomorphic cubic fields $L_1,\ldots,L_m$ sharing a common discriminant $d(L_i) = d(L)$ with $L$ is determined.

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arXiv2021

New perspectives of the power-commutator-structure: Coclass trees of CF-groups and related BCF-groups

Let e>1 be an integer. Among the finite 3-groups G with bicyclic commutator quotient G/G' ~ C(3^e) * C(3), having one non-elementary component with logarithmic exponent e, there exists a unique pair of coclass trees with distinguished rank distribution rho ~ (2,2,3;3). One tree T(e)(M(e,1)) consists of CF-groups with coclass e, and the other tree T(e+1)(M(e+1,1)) consists of BCF-groups with coclass e+1. It is proved that, due to a chain of periodic bifurcations, the vertices of all pairs (T(e),T(e+1)) with e>2 can be constructed as p-descendants of the single root M(3,1) of order 729 by means of the p-group generation algorithm by Newman and O'Brien.

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arXiv2021

Periodic Schur sigma-groups of non-elementary bicyclic type

Infinitely many large Schur sigma-groups G with non-elementary bicyclic commutator quotient G/G' = C(3^e) x C(3), e >= 2, are constructed as periodic sequences of vertices in descendant trees of finite 3-groups. A single root gives rise to pairs of metabelian groups G with logarithmic order lo(G) = 4+e for e >= 3. Three roots are ancestors of pairs of non-metabelian groups G with moderate rank distribution rho(G) = (2,2,3;3) and lo(G) = 7+e for e >= 5. Twentyseven roots produce sextets of non-metabelian groups G with elevated rank distribution rho(G) = (3,3,3;3) and lo(G) = 19+e for e >= 9. The soluble length of non-metabelian groups is always sl(G) = 3. The groups can be realized as 3-class field tower groups Gal(F(3,infty,K)/K) of imaginary quadratic number fields K = Q(d^1/2) with fundamental discriminants d < 0.

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arXiv2021

First excited state with moderate rank distribution

Evidence is provided for the existence of infinite periodic sequences of Schur sigma-groups G with commutator quotient G/G' ~ C(3^e) x C(3), e >= 7, and logarithmic order lo(G) = 10+e. With respect to their maximal subgroups H1,H2,H3;H4, they have moderate rank distribution rho(G) = (rank3(Hi/Hi')) ~ (2,2,3;3) and represent the first excited state of their punctured transfer kernel types kappa(G), which is characterized by a polarized component, e33 or (e+1)32, of the abelian quotient invariants alpha(G) = (Hi/Hi') with lo = 6+e in contrast to the ground state with lo = 4+e.

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arXiv2021

BCF-groups with elevated rank distribution

Infinitely many large Schur sigma-groups G with logarithmic order lo(G)=19+e, non-elementary bicyclic commutator quotient G/G' ~ C(3^e) x C(3), e >= 2, elevated rank distribution rho(G)=(3,3,3;3), punctured transfer kernel type kappa(G) ~ (144;4) and soluble length sl(G)=3 are constructed. Up to e <= 4, they are realized as 3-class field tower groups Gal(F(3,infty,K)/K) of imaginary quadratic number fields K=Q(d^1/2), d<0. Their metabelianizations M=G/G'' are BCF-groups with lo(M)=8+e and bicyclic third lower central factor gamma3(M)/gamma4(M) ~ C(3) x C(3).

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arXiv2021

Bicyclic commutator quotients with one non-elementary component

For any number field K with non-elementary 3-class group Cl(3,K) = C(3^e) x C(3), e >= 2, the punctured capitulation type kappa(K) of K in its unramified cyclic cubic extensions Li, 1 <= i <= 4, is an orbit under the action of S3 x S3. By means of Artin's reciprocity law, the arithmetical invariant kappa(K) is translated to the punctured transfer kernel type kappa(G2) of the automorphism group G2 = Gal(F(3,2,K)/K) of the second Hilbert 3-class field of K. A classification of finite 3-groups G with low order and bicyclic commutator quotient G/G' = C(3^e) x C(3), 2 <= e <= 6, according to the algebraic invariant kappa(G), admits conclusions concerning the length of the Hilbert 3-class field tower F(3,infty,K) of imaginary quadratic number fields K.

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arXiv2021

$3$-Principalization over $S_3$-fields

Let $p\equiv 1\,(\mathrm{mod}\,9)$ be a prime number and $ζ_3$ be a primitive cube root of unity. Then $\mathrm{k}=\mathbb{Q}(\sqrt[3]{p},ζ_3)$ is a pure metacyclic field with group $\mathrm{Gal}(\mathrm{k}/\mathbb{Q})\simeq S_3$. In the case that $\mathrm{k}$ possesses a $3$-class group $C_{\mathrm{k},3}$ of type $(9,3)$, the capitulation of $3$-ideal classes of $\mathrm{k}$ in its unramified cyclic cubic extensions is determined, and conclusions concerning the maximal unramified pro-$3$-extension $\mathrm{k}_3^{(\infty)}$ of $\mathrm{k}$ are drawn.

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arXiv2021

Classifying multiplets of totally real cubic fields

The number of non-isomorphic cubic fields L sharing a common discriminant d(L) = d is called the multiplicity m = m(d) of d. For an assigned value of d, these fields are collected in a multiplet M(d) = (L(1) ,..., L(m)). In this paper, the information in all existing tables of totally real cubic number fields L with positive discriminants d(L) < 10000000 is extended by computing the differential principal factorization types tau(L) in (alpha1, alpha2, alpha3, beta1, beta2, gamma, delta1, delta2, epsilon) of the members L of each multiplet M(d) of non-cyclic fields, 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 (U(N) : U(0)) = 1 for ramified extensions N/K with conductor f > 1 is refined and verified.

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