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We generalize the Chebotarev density formulas of Dawsey (2017) and Alladi (1977) to the setting of arbitrary finite Galois extensions of number fields $L/K$. In particular, if $C \subset G = \textrm{Gal}(L/K)$ is a conjugacy class, then we…

A connected reductive group G over a field k may be written as a quotient H/S, where the k-group H is an extension of a quasitrivial torus by a simply connected semisimple group, and S is a flasque k-torus, central in H (a flasque torus is…

In his previous papers (Math. Res. Letters 7 (2000), 123--13; Math. Res. Letters 8 (2001), 429--435; Moscow Math. J. 2 (2002), issue 2, 403-431) the author proved that in characteristic $\ne 2$ the jacobian $J(C)$ of a hyperelliptic curve…

Given a number field $F$, a finite group $G$ and an indeterminate $T$, {\it{a $G$-parametric extension over $F$}} is a finite Galois extension $E/F(T)$ with Galois group $G$ and $E/F$ regular that has all the Galois extensions of $F$ with…

For a linear differential equation defined over a formally real differential field K with real closed field of constants k, Crespo, Hajto and van der Put proved that there exists a unique formally real Picard- Vessiot extension up to…

The fundamental concepts in the Galois Theory are separable, normal and Galois field extensions. These concepts are central in proofs of the Galois Theory. In the paper, we introduce a new approach, a ring theoretic approach, to the Galois…

In this note, we are interested in local-global principles for multinorm equations of the form $\prod_{i=1}^n N_{L_i /k}(z_i) = a$ where $k$ is a global field, $L_i/k$ are finite separable field extensions and $a \in k^*$. In particular, we…

We show that for any given field $k$ and natural number $r\geq2$, every continuous extension of the absolute Galois group $\mathrm{Gal}_k$ by a finite group is the arithmetic fundamental group of a geometrically connected smooth projective…

In this paper we consider the problem of classifying the isomorphism classes of extensions of degree pk of a p-adic field, restricting to the case of extensions without intermediate fields. We establish a correspondence between the…

The work of Greither and Pareigis details the enumeration of the Hopf-Galois structures (if any) on a given separable field extension. For an extension $L/K$ which is classically Galois with $G=Gal(L/K)$ the Hopf algebras in question are of…

Let $G$ be a finite group. Let $K/k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C \subseteq \mathrm{Gal}(K/k) \simeq G$ be a conjugacy invariant subset. It is well known that there exists an…

We show that the Andr\'{e} motive of a hyper-K\"{a}hler variety $X$ over a field $K \subset \mathbb{C}$ with $b_2(X)>6$ is governed by its component in degree $2$. More precisely, we prove that if $X_1$ and $X_2$ are deformation equivalent…

This paper proves local-global principles for Galois cohomology groups over function fields $F$ of curves that are defined over a complete discretely valued field. We show in particular that such principles hold for $H^n(F, Z/mZ(n-1))$, for…

Let k be a number field and K/k Galois. We transform the construction of the unramified Brauer group of the norm one torus R^1_K/k(G_m) into the construction of a special abelian extension over K. If k=Q and K/Q biquadratic, we explicitly…

In his thesis, S. Checcoli shows that, among other results, if $K$ is a number field and if $L/K$ is an infinite Galois extension with Galois group $G$ of finite exponent, then $L$ has uniformly bounded local degrees at every prime of $K$.…

Let $k$ be a finitely generated field of characteristic $p>0$ and $X$ a smooth and proper scheme over $k$. Recent works of Cadoret, Hui and Tamagawa show that, if $X$ satisfies the $\ell$-adic Tate conjecture for divisors for every prime…

We consider Hopf Galois structures on a separable field extension $L/K$ of degree $p^n$, for $p$ an odd prime number, $n\geq 3$. For $p > n$, we prove that $L/K$ has at most one abelian type of Hopf Galois structures. For a nonabelian group…

For a prime number $\ell$ and an extension of number fields $K/F$, we prove new lower bounds on the $\ell$-rank of the ideal class group of $K$ based on prime ramification in $K/F$. Unlike related results from the literature, our bound is…

The paper is concerned with the following version of Hilbert's irreducibility theorem: if $\pi: X \to Y$ is a Galois $G$-covering of varieties over a number field $k$ and $H \subset G$ is a subgroup, then for all sufficiently large and…

We prove three theorems concerning the Hopf-Galois module structure of fractional ideals in a finite tamely ramified extension of $ p $-adic fields or number fields which is $ H $-Galois for a commutative Hopf algebra $ H $. Firstly, we…