Related papers: Lower bounds on the $\ell$-rank of ideal class gro…

Let $K/F$ be a finite extension of number fields of degree $n \geq 2$. We establish effective field-uniform unconditional upper bounds for the least norm of a prime ideal of $F$ which is degree 1 over $\mathbb{Q}$ and does not ramify or…

Let $G$ be a finite nilpotent group and $K$ a number field with torsion relatively prime to the order of $G$. By a sequence of central group extensions with cyclic kernel we obtain an upper bound for the minimum number of prime ideals of…

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…

Given a $p$-adic field $K$ and a prime number $\ell$, we count the total number of the isomorphism classes of $p^\ell$-extensions of $K$ having no intermediate fields. Moreover for each group that can appear as Galois group of the normal…

We prove a local-global principle for the embedding problems of global fields with restricted ramification. By this local-global principle, for a global field $k$, we use only the local information to give a presentation of the maximal…

Let T be a free Z_p-module of finite rank equipped with a continuous Z_p-linear action of the absolute Galois group of a number field K satisfying certain conditions. In this article, by using a Selmer group corresponding to T, we give a…

We investigate unramified extensions of number fields with prescribed solvable Galois group and certain extra conditions. In particular, we are interested in the minimal degree of a number field $K$, Galois over $\mathbb{Q}$, such that $K$…

In this paper, we consider infinite Galois extensions of number fields and study the relation between their local degrees and the structure of their Galois groups. It is known that, if $K$ is a number field and $L/K$ is an infinite Galois…

In this work, we show that given a finite p-group G, a number field K having a trivial p-class group Cl K , and a finite set of primes S of K, there exists a finite extension F/K such that the S-split p-Hilbert class field tower L S p (F )…

Let $p$ be an odd prime number. We study growth patterns associated with finitely ramified Galois groups considered over the various number fields varying in a $\mathbb{Z}_p$-tower. These Galois groups can be considered as non-commutative…

For all positive integers $\ell$, we prove non-trivial bounds for the $\ell$-torsion in the class group of $K$, which hold for almost all number fields $K$ in certain families of cyclic extensions of arbitrarily large degree. In particular,…

Let $A$ be a finite commutative nilpotent $\mathbb{F}_p$-algebra structure on $G$, an elementary abelian group of order $p^n$. If $K/k$ is a Galois extension of fields with Galois group $G$ and $A^p = 0$, then corresponding to $A$ is an…

We show that if F is the rational numbers or a multiquadratic number field, p is 2,3, or 5, and K/F is a Galois extension of degree a power of p, then for elliptic curves E/Q ordered by height, the average dimension of the p-Selmer groups…

Let $G$ be a Frobenius group with an abelian Frobenius kernel $F$ and let $k$ be a finite extension of $\mathbb{Q}$. We obtain an upper bound for the number of degree $|F|$ algebraic extensions $K/k$ with Galois group $G$ with the norm of…

Let K and F be complete discrete valuation fields of residue characteristic p>0. Let m be a positive integer no more than their absolute ramification indices. Let s and t be their uniformizers. Let L/K and E/F be finite extensions such that…

Let $K$ be a local field of characteristic 0 with residue characteristic $p$. Let $G$ be an extraspecial $p$-group and let $L/K$ be a totally ramified $G$-extension. In this paper we find sufficient conditions for $L/K$ to admit a Galois…

Let $\ell\geq 5$ be a prime number and $\mathbb{F}_\ell$ denote the finite field with $\ell$ elements. We show that the number of Galois extensions of the rationals with Galois group isomorphic to $GL_2(\mathbb{F}_\ell)$ and absolute…

Given a natural number n and a number field K, we show the existence of an integer \ell_0 such that for any prime number \ell\geq \ell_0, there exists a finite extension F/K, unramified in all places above \ell, together with a principally…

Let A be an abelian variety defined over a number field k and F a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we investigate the explicit Galois structure of the…

For each finite subgroup $G$ of $PGL_2(\mathbb{Q})$, and for each integer $n$ coprime to $6$, we construct explicitly infinitely many Galois extensions of $\mathbb{Q}$ with group $G$ and whose ideal class group has $n$-rank at least…