Related papers: Three stage towers of 5-class fields

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…

Let zeta5 be a primitive fifth root of unity and d<>1 be a quadratic fundamental discriminant not divisible by 5. For the 5-dual cyclic quartic field M=Q((zeta5-zeta5^-1)*d^1/2) of the quadratic fields k1=Q(d^1/2) and k2=Q((5*d)^1/2) in the…

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…

The $p$-group generation algorithm is used to verify that the Hilbert $3$-class field tower has length $3$ for certain imaginary quadratic fields $K$ with $3$-class group $\mathrm{Cl}_3(K) \cong [3,3]$. Our results provide the first…

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…

Let $G$ be a $5$-group of maximal class and $\gamma_2(G) = [G, G]$ its derived group. Assume that the abelianization $G/\gamma_2(G)$ is of type $(5, 5)$ and the transfers $V_{H_1\to \gamma_2(G)}$ and $V_{H_2\to \gamma_2(G)}$ are trivial,…

For a fixed prime p, the p-class tower F(p,infinity,K) of a number field K is considered to be known if a pro-p presentation of the Galois group H = Gal( F(p,infinity,K)/K ) is given. In the last few years, it turned out that the Artin…

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…

Let p be a prime and K be a number field with non-trivial p-class group Cl(p,K). A crucial step in identifying the Galois group G=G(p,K) of the maximal unramified pro-p extension of K is to determine its two-stage approximation M=G(p,2,K),…

The p-class tower $F_p^\infty(k)$ of a number field k is its maximal unramified pro-p extension. It is considered to be known when the p-tower group, that is the Galois group $G:=Gal(F_p^\infty(k)/k)$, can be identified by an explicit…

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…

Let F be a number field and p be a prime. In the Successive Approximation Theorem, we prove that, for each positive integer n, finitely many candidates for the Galois group G(p,n,F) of the n-th stage F(p,n) of the p-class tower…

The group of isometries W of a regular rooted tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in W. This fact has stimulated the computation of the group of automorphisms of such…

Recent examples of periodic bifurcations in descendant trees of finite p-groups with p in {2,3} are used to show that the possible p-class tower groups G of certain multiquadratic fields K with p-class group of type (2,2,2), resp. (3,3),…

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) ~…

Let p be a prime. For any finite p-group G, the deep transfers T(H,G'):H/H' --> G'/G'' from the maximal subgroups H of index (G:H)=p in G to the derived subgroup G' are introduced as an innovative tool for identifying G uniquely by means of…

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…

In this article we continue the investigation of the length of the narrow $2$-class field tower of real quadratic number fields $\mathrm{k}$ whose discriminants are not a sum of two squares and for which their $2$-class groups are…

This paper studies infinite class field towers of number fields $K$ that are ramified over $\Q$ only at one finite prime. In particular, we show the existence of such towers for a general family of primes including $p=2$, 3 and 5.

We prove the existence of infinitely many real and imaginary fields whose 5-rank of the class group is >=3.