Related papers: Analysis and Probability over Infinite Extensions …

Given a number field $k$, we show that, for many finite groups $G$, all the Galois extensions of $k$ with Galois group $G$ cannot be obtained by specializing any given finitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$…

We consider the distribution of the Galois groups $\operatorname{Gal}(K^{\operatorname{un}}/K)$ of maximal unramified extensions as $K$ ranges over $\Gamma$-extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$. We prove two properties of…

For a particular class of Galois structures, we prove that the normal extensions are precisely those extensions that are "locally" split epic and trivial, and we use this to prove a "Galois theorem" for normal extensions. Furthermore, we…

We study the relationship between the local and global Galois theory of function fields over a complete discretely valued field. We give necessary and sufficient conditions for local separable extensions to descend to global extensions, and…

We prove that if $K/\mathbb{Q}$ is a Galois extension of finite exponent and $K^{(d)}$ is the compositum of all extensions of $K$ of degree at most $d$, then $K^{(d)}$ has the Bogomolov property and the maximal abelian subextension of…

Let K be a local field of characteristic p with perfect residue field k. In this paper we find a set of representatives for the k-isomorphism classes of totally ramified separable extensions L/K of degree p. This extends work of Klopsch,…

We generalize Sen theory to extensions $K_\infty/K$ whose Galois group is a $p$-adic Lie group of arbitrary dimension. To do so, we replace Sen's space of $K$-finite vectors by Schneider and Teitelbaum's space of locally analytic vectors.…

Let $G$ be an algebraic group, $X$ a generically free $G$-variety, and $K=k(X)^G$. A field extension $L$ of $K$ is called a splitting field of $X$ if the image of the class of $X$ under the natural map $H^1(K, G) \mapsto H^1(L, G)$ is…

For a finite Galois extension K/k and an intermediate field F such that Gal(K/F) has a normal complement in Gal(K/k), we construct and characterize Hopf Galois structures on K/k which are induced by a pair of Hopf Galois structures on K/F…

We provide a characterization of infinite algebraic Galois extensions of the rationals with uniformly bounded local degrees, giving a detailed proof of all the results announced in a paper by Checcoli and Zannier and obtaining relevant…

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…

The theory of general Galois-type extensions is presented, including the interrelations between coalgebra extensions and algebra (co)extensions, properties of corresponding (co)translation maps, and rudiments of entwinings and…

In this note, we compute the {\Sigma}^1(G) invariant when 1 {\to} H {\to} G {\to} K {\to} 1 is a short exact sequence of finitely generated groups with K finite. As an application, we construct a group F semidirect Z_2 where F is the R.…

Let $K$ be a number field and $S$ a set of primes of $K$. We write $K_S/K$ for the maximal extension of $K$ unramified outside $S$ and $G_{K,S}$ for its Galois group. In this paper, we answer the following question under some assumptions:…

For all simple and finite extension of a valued field, we prove that its defect is the product of the effective degrees of the complete set of key polynomials associated. As a consequence, we obtain a local uniformization theorem for…

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.…

Let K be a complete discretely valued field with residue field k and F be a function field of a curve over K. Let L/F be a Galois extension of degree n. If n is coprime to char(k), then under some assumptions on k(e.g. k is algebraically…

Let $K$ be a finite tamely ramified extension of $\Q_p$ and let $L/K$ be a totally ramified $(\Z/p^n\Z)$-extension. Let $\pi_L$ be a uniformizer for $L$, let $\sigma$ be a generator for $\Gal(L/K)$, and let $f(X)$ be an element of $\O_K[X]$…

The aim of the paper is to introduce B-extensions which are the most symmetrical finite field extensions (a finite field extension $L/K$ is called a {\it B-extension} if the endomorphism algebra ${\rm End}_K(L)$ is generated by the algebra…

Let K be a finite extension of Q_p with residue field F_q and let P(T) = T^d + a_{d-1}T^{d-1} + ... +a_1 T, where d is a power of q and a_i is in the maximal ideal of K for all i. Let u_0 be a uniformizer of O_K and let {u_n}_{n \geq 0} be…