Daniel C. Mayer
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View article: The Capitulation Problem in Certain Pure Cubic Fields
The Capitulation Problem in Certain Pure Cubic Fields Open
Let \(Γ=\mathbb{Q}(\sqrt[3]{n})\) be a pure cubic field with normal closure \(k=\mathbb{Q}(\sqrt[3]{n},ζ)\), where \(n>1\) denotes a cube free integer, and \(ζ\) is a primitive cube root of unity. Suppose \(k\) possesses an elementary bicy…
View article: Group theory of cyclic cubic number fields
Group theory of cyclic cubic number fields Open
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-…
View article: A Group Theoretic Approach to Cyclic Cubic Fields
A Group Theoretic Approach to Cyclic Cubic Fields Open
Let (kμ)μ=14 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 of the quartet whose 3-class group Cl3(kμ)≃(Z/3Z)2 is elementary bicyclic, the a…
View article: Group theoretic approach to cyclic cubic fields
Group theoretic approach to cyclic cubic fields Open
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 bic…
View article: Cyclic cubic number fields with harmonically balanced capitulation
Cyclic cubic number fields with harmonically balanced capitulation Open
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 rel…
View article: Theoretical and Experimental Approach to p-Class Field Towers of Cyclic Cubic Number Fields
Theoretical and Experimental Approach to p-Class Field Towers of Cyclic Cubic Number Fields Open
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 …
View article: Schur sigma-Groups of Scholz-Taussky Type F
Schur sigma-Groups of Scholz-Taussky Type F Open
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. Evidenc…
View article: Bicyclic commutator quotients with one non-elementary component
Bicyclic commutator quotients with one non-elementary component Open
For any number field $K$ with non-elementary $3$-class group ${\rm Cl}_3(K)\simeq C_{3^e}\times C_3$, $e\ge2$, the punctured capitulation type $\varkappa(K)$ of $K$ in its unramified cyclic cubic extensions $L_i$, $1\le i\le4$, is an orbit…
View article: Algebraic number fields generated by an infinite family of monogenic trinomials
Algebraic number fields generated by an infinite family of monogenic trinomials Open
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 =…
View article: New perspectives of the power-commutator-structure: Coclass trees of CF-groups and related BCF-groups
New perspectives of the power-commutator-structure: Coclass trees of CF-groups and related BCF-groups Open
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…
View article: Periodic Schur sigma-groups of non-elementary bicyclic type
Periodic Schur sigma-groups of non-elementary bicyclic type Open
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…
View article: First excited state with moderate rank distribution
First excited state with moderate rank distribution Open
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, t…
View article: BCF-groups with elevated rank distribution
BCF-groups with elevated rank distribution Open
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) ~ …
View article: Bicyclic commutator quotients with one non-elementary component
Bicyclic commutator quotients with one non-elementary component Open
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 …
View article: Classifying multiplets of totally real cubic fields
Classifying multiplets of totally real cubic fields Open
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 n…
View article: Construction and classification of \(p\)-ring class fields modulo \(p\)-admissible conductors
Construction and classification of \(p\)-ring class fields modulo \(p\)-admissible conductors Open
Each \(p\)-ring class field \(K_f\) modulo a \(p\)-admissible conductor \(f\) over a quadratic base field \(K\) with \(p\)-ring class rank \(\varrho_f\) mod \(f\) is classified according to Galois cohomology and differential principal fact…
View article: Classifying multiplets of totally real cubic fields
Classifying multiplets of totally real cubic fields Open
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…
View article: Construction and classification of p-ring class fields modulo p-admissible conductors
Construction and classification of p-ring class fields modulo p-admissible conductors Open
Each p-ring class field K(f) modulo a p-admissible conductor f over a quadratic base field K with p-ring class rank r(f) mod f is classified according to Galois cohomology and differential principal factorization type of all members of its…
View article: Finite non-metabelian Schur sigma-Galois groups of class field towers
Finite non-metabelian Schur sigma-Galois groups of class field towers Open
For each odd prime p>=5, there exist finite p-groups G with derived quotient G/D(G)=C(p)xC(p) and nearly constant transfer kernel type k(G)=(1,2,...,2) having two fixed points. It is proved that, for p=7, this type k(G) with the simplest p…
View article: Schur sigma-groups with abelian quotient invariants (9,3)
Schur sigma-groups with abelian quotient invariants (9,3) Open
By the construction of suitable non-metabelian Schur sigma-groups S of type (9,3) with log order lo(S) = 21 and nilpotency class cl(S) = 9, evidence is provided of a new class of imaginary quadratic fields K with 3-class group Cl(3,K) ~ C(…
View article: Extremal root paths of Schur \(\sigma\)-groups and first \(3\)-class field towers with four stages.
Extremal root paths of Schur \(\sigma\)-groups and first \(3\)-class field towers with four stages. Open
An extremal property of finite Schur sigma-groups G is described in terms of their path to the root in the descendant tree of their abelianization G/G'. The phenomenon is illustrated and verified by all known examples of Galois groups G=Ga…
View article: Extremal root paths of Schur \(σ\)-groups and first \(3\)-class field towers with four stages
Extremal root paths of Schur \(σ\)-groups and first \(3\)-class field towers with four stages Open
An extremal property of finite Schur sigma-groups G is described in terms of their path to the root in the descendant tree of their abelianization G/G'. The phenomenon is illustrated and verified by all known examples of Galois groups G=Ga…
View article: Harmonically balanced capitulation over quadratic fields of type (9,9)
Harmonically balanced capitulation over quadratic fields of type (9,9) Open
The isomorphism type of the Galois group G of finite 3-class field towers of quadratic number fields with 3-class group of type (9,9) is determined by means of Artin patterns which contain information on the transfer of 3-classes to unrami…
View article: Principal factors and lattice minima
Principal factors and lattice minima Open
Let $\mathit{k}=\mathbb{Q}(\sqrt[3]{d},ζ_3)$, where $d>1$ is a cube-free positive integer, $\mathit{k}_0=\mathbb{Q}(ζ_3)$ be the cyclotomic field containing a primitive cube root of unity $ζ_3$, and $G=\operatorname{Gal}(\mathit{k}/\mathit…
View article: The strategy of pattern recognition via Artin transfers applied to finite towers of 2-class fields
The strategy of pattern recognition via Artin transfers applied to finite towers of 2-class fields Open
The isomorphism type of the Galois group of the 2-class field tower of quadratic number fields having a 2-class group with abelian type invariants (4,4) is determined by means of information on the transfer of 2-classes to unramified abeli…
View article: Differential principal factors and Pólya property of pure metacyclic fields
Differential principal factors and Pólya property of pure metacyclic fields Open
Barrucand and Cohn’s theory of principal factorizations in pure cubic fields [Formula: see text] and their Galois closures [Formula: see text] with [Formula: see text] types is generalized to pure quintic fields [Formula: see text] and pur…
View article: Generalized Artin pattern of heterogeneous multiplets of dihedral fields and proof of Scholz's conjecture
Generalized Artin pattern of heterogeneous multiplets of dihedral fields and proof of Scholz's conjecture Open
The concept of Artin transfer pattern $((\ker(T_{K,N_i}))_i,(\mathrm{Cl}_p(N_i))_i)$ for homogeneous multiplets $(N_1,\ldots,N_m)$ of unramified cyclic prime degree p extensions $N_i/K$ of a base field K with p-class transfer homomorphisms…
View article: Tables of Pure Quintic Fields
Tables of Pure Quintic Fields Open
By making use of our generalization of Barrucand and Cohn's theory of principal factorizations in pure cubic fields $\mathbb{Q}(\sqrt[3]{D})$ and their Galois closures $\mathbb{Q}(ζ_3,\sqrt[3]{D})$ with 3 possible types to pure quintic fie…