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Suppose that $K$ is a field of characteristic 0, $p$ is an odd prime, $r$ a positive integer, $q=p^r$ a prime power. Suppose that $f(x)$ is a polynomial of degree $n > 4$ with coefficients in $K$ and without multiple roots. Let us consider…

This paper proves that if $E$ is a field, such that the Galois group $\mathcal{G}(E(p)/E)$ of the maximal $p$-extension $E(p)/E$ is a Demushkin group of finite rank $r(p)_{E} \ge 3$, for some prime number $p$, then $\mathcal{G}(E(p)/E)$…

Motivated by the work of Liu, we study certain canonical quotients of $G_{\emptyset}^T(K)$ -- the Galois group of the maximal unramified extension of a global field $K$ that is split completely at a finite nonempty set of places in $T$ --…

Let $k\subseteq K$ be a finite Galois extension of fields with Galois group $G$. Let $\mathscr{G}$ be the automorphism $k$-group scheme of $K$. We construct a canonical $k$-subgroup scheme $\underline{G}\subset\mathscr{G}$ with the property…

Let $L/K$ be a finite Galois, totally ramified $p$-extension of complete local fields with perfect residue fields of characteristic $p>0$. In this paper, we give conditions, valid for any Galois $p$-group $G={Gal}(L/K)$ (abelian or not) and…

We compute the Z-rank of the subgroup of elements of the multiplicative group of a number field K that are norms from every finite level of the cyclotomic Z{\ell}-extension of K. Thus we compare its {\ell}-adification with the group of…

For a finite extension $K/\mathbb{Q}_p$ and a split reductive group $G$ over $\mathcal{O}_K$, let $\overline{\rho} \colon \mathrm{Gal}_K \to G(\overline{\mathbb{F}}_p)$ be a continuous quasi-semisimple mod $p$ $G$-valued representation of…

For primes $p$ and $\ell$ such that $\ell$ divides $p-1$, Hirano and Morishita constructed a nonabelian Galois extension of the function field $\mathbb{F}_p(t)$ whose degree is $\ell^3$ and Galois group is of Heisenberg type. Here we…

The power classes of a field are well-known for their ability to parameterize elementary $p$-abelian Galois extensions. These classical objects have recently been reexamined through the lens of their Galois module structure. Module…

For a maximal separable subfield $K$ of a central simple algebra $A$, we provide a semiring isomorphism between $K$-$K$-bimodules $A$ and $H$-$H$ bisets of $G = \Gal(L/F)$, where $F = \operatorname{Z}(A)$, $L$ is the Galois closure of…

Let $ K$ be a field admitting a Galois extension $L$ of degree $n$, denoting the Galois group as $G = \gal(L/K)$. Our focus lies on the space $\sym_K(L)$ of symmetric $K$-bilinear forms on $L$. We establish a decomposition of $\sym_K(L)$…

For a number field $K$, we consider $K^{\rm ta}$ the maximal tamely ramified algebraic extension of~$K$, and its Galois group $G^{\rm ta}_K= Gal(K^{ta}/K)$. Choose a prime $p$ such that $\mu_p \not \subset K$. Our guiding aim is to…

For a wildly ramified extension $K/k$ of complete discrete valuation fields we study collections of elements of $k[G]$ (where $G=Gal(K/k)$) that fit well for constructing bases of various associated Galois modules and orders. In the case…

Let $F/E$ be a Galois extension of totally real number fields, with Galois group $\mathrm{Gal}(F/E)$. Let $\mathfrak{N}$ be an integral ideal which is $\mathrm{Gal}(F/E)$-invariant, and $k \ge 2$ an integer. In this note, we study the…

Let $\mathcal{P}: \cdots \rightarrow C_2\rightarrow C_1\rightarrow {\mathbb P}^1$ be a $\mathbb{Z}_p$-cover of the projective line over a finite field of cardinality $q$ and characteristic $p$ which ramifies at exactly one rational point,…

Let $K$ be a number field, and let $G\subset K^\times$ be a finitely generated subgroup. Fix some prime number $\ell$, and consider the set of primes $\mathfrak{p}$ of $K$ satisfying the following property: the reduction of $G$ modulo…

The aim of this paper is to investigate the algebraicity behavior of reductions of $D$-finite power series modulo prime numbers. For many classes of D-finite functions, such as diagonals of multivariate algebraic series or hypergeometric…

Fix an abelian variety $A$ of dimension $g\geq 1$ defined over a number field $K$. For each prime $\ell$, the Galois action on the $\ell$-power torsion points of $A$ induces a representation $\rho_{A,\ell}\colon Gal_K \to…

Let $\ell \geq 5$ be a prime and let $N$ be a square-free integer prime to $\ell$. For each prime $p$ dividing $N$, let $a_p$ be either $1$ or $-1$. We give sufficient criteria for the existence of a newform $f$ of weight 2 for…

Let K be a field admitting a cyclic Galois extension of degree n. The main result of this paper is a decomposition theorem for the space of alternating bilinear forms defined on a vector space of odd dimension n over K. We show that this…