Related papers: Infinite class field towers

In this paper we study general conditions to prove the infiniteness of the genus of certain towers of function fields over a perfect field. We show that many known examples of towers with infinite genus are particular cases of these…

Over all non-prime finite fields, we construct some recursive towers of function fields with many rational places. Thus we obtain a substantial improvement on all known lower bounds for Ihara's quantity $A(\ell)$, for $\ell = p^n$ with $p$…

Let $K$ be a cyclic totally real number field of odd degree over $\mathbb{Q}$ with odd class number, such that every totally positive unit is the square of a unit, and such that $2$ is inert in $K/\mathbb{Q}$. We define a family of number…

Given a prime $p$, a number field $\K$ and a finite set of places $S$ of $\K$, let $\K_S$ be the maximal pro-$p$ extension of $\K$ unramified outside $S$. Using the Golod-Shafarevich criterion one can often show that $\K_S/\K$ is infinite.…

We use the notion of an Etesi $C^*$-algebra to prove that the real class field towers are always finite.

To extend Iwasawa's classical theorem from ${\mathbb Z}_p$-towers to ${\mathbb Z}_p^d$-towers, Greenberg conjectured that the exponent of $p$ in the $n$-th class number in a ${\mathbb Z}_p^d$-tower of a global field $K$ ramified at finitely…

We determine some properties of the narrow 2-class field tower of those real quadratic number fields whose discriminants are not a sum of two squares and for which their 2-class groups are elementary of order $4$. Here in Part I, we…

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…

The structure of the Galois group of the maximal unramified p-extension of an imaginary quadratic field is restricted in various ways. In this paper we construct a family of finite 3-groups satisfying these restrictions. We prove several…

In a previous work general conditions were given to prove the infiniteness of the genus of certain towers of function fields over a perfect field. It was shown that many examples where particular cases of those general results. In this…

Let $K/\Q$ be a cyclic extension of number fields with Galois group $G$. We study the ideal classes of primes $\mathfrak{p}$ of $K$ of residue degree bigger than one in the class group of $K$. In particular, we explore such extensions…

Let k be a number field, p$\ge$2 a prime and S a set of tame or wild finite places of k. We call K/k a totally S-ramified cyclic p-tower if Gal(K/k)=Z/p^NZ and if S non-empty is totally ramified. Using analogues of Chevalley's formula…

Let $p$ be an odd prime and $F$ be a number field whose $p$-class group is cyclic. Let $F_{\{\mathfrak{q}\}}$ be the maximal pro-$p$ extension of $F$ which is unramified outside a single non-$p$-adic prime ideal $\mathfrak{q}$ of $F$. In…

For any odd prime $p,$ we construct an infinite family of pairs of imaginary quadratic fields $\mathbb{Q}(\sqrt{d}),\mathbb{Q}(\sqrt{d+1})$ whose class numbers are both divisible by $p.$ One of our theorems settles Iizuka's conjecture for…

For a prime p, we study the Galois groups of maximal pro-$p$ extensions of imaginary quadratic fields unramified outside a finite set $S$, where $S$ consists of one or two finite places not lying above $p$. When $p$ is odd, we give explicit…

In this paper, we investigate the unit groups, the $2$-class groups, the $2$-class field towers and the structures of the second $2$-class groups of some multiquadratic number fields of degree $8$ and $16$.

We determine the Galois group of the 2-class field tower for two particular families of imaginary quadratic number fields $k$ with $2$-class field tower of length $2$.

Fix a finite collection of primes $\{ p_j \}$, not containing $2$ or $3$. Using some observations which arose from attempts to solve the SIC-POVMs problem in quantum information, we give a simple methodology for constructing an infinite…

We answer a question of Peikert and Rosen by giving for each $\epsilon > 0$ an efficient construction of infinite families of number fields $N$ such that the root discriminant $D_N^{1/[N:\mathbb{Q}]}$ is bounded above by a constant times…

With K=Q((3812377)^(1/2)) we give the first example of an algebraic number field possessing a 5-class tower of exact length L(5,K)=3. The rigorous proof is conducted by means of the p-group generation algorithm, showing the existence of a…