Related papers: Norm principle for even K-groups of number fields

In this note we generalize Nori's definition of the fundamental group scheme from a rational point to an arbitrary base point so that when we take $X$ to be a field $k$ and the point to be $k\subseteq \bar{k}$ we still get a non trivial…

Let $q$ be a prime power of a prime $p$, $n$ a positive integer and $\mathbb F_{q^n}$ the finite field with $q^n$ elements. The $k-$normal elements over finite fields were introduced and characterized by Huczynska et al (2013). Under the…

We consider from a geometric point of view the conjectural fundamental lemma of Langlands and Shelstad for unitary groups over a local field of positive characteristic. We introduce projective algebraic varieties over the finite residue…

We prove the version of Knebusch's Norm principle for simple extensions of (semi-)local rings. As an application we prove the Grothedieck-Serre's conjecture on principal homogeneous spaces for the split case of the spinor group.

We estimate the distribution of relatively $r$-prime lattice points in number fields $K$ with their components having a norm less than $x$. In the previous paper we obtained uniform upper bounds as $K$ runs through all number fields under…

We give an analytic proof of the norm index theorem $[I_:K^* N(I_L)] =[L:K]$ for cyclic extensions of number fields using spectral theory of the idele class group.

In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the…

We prove in this article the surjectivity of three maps. We prove in Theorem $1.6$ the surjectivity of the Chinese remainder reduction map associated to the projective space of an ideal with a given factorization into ideals whose radicals…

Polynomials and elements over finite fields exhibit closely related algebraic structures, and many properties defined for elements extend naturally to polynomials. The concepts of order and $\mathbb{F}_q$-Order for elements have been…

We introduce and study some families of groups whose irreducible characters take values on quadratic extensions of the rationals. We focus mostly on a generalization of inverse semi-rational groups, which we call uniformly semi-rational…

In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of $k$-normal elements: an element $\alpha \in \mathbb{F}_{q^n}$ is $k$-normal over $\mathbb{F}_q$ if the greatest common divisor of the polynomials $g_{\alpha}(x)=…

Let K be an arbitrary field. We will determine explicitly all the nontrivial finite groups of essential dimension one over K.

Given a number field $k$ and a finite $k$-group $G$, the Tame Approximation Problem for $G$ asks whether the restriction map $H^1(k,G)\to\prod_{v\in\Sigma}H^1(k_v,G)$ is surjective for every finite set of places $\Sigma\subseteq\Omega_k$…

A ``k-rule" is a sequence A=((A_n,B_n):n<omega) of pairwise disjoint sets B_n, each of cardinality at most k, where A_n is a subset of B_n. A set X of natural numbers (a ``real'') follows a rule A if for infinitely many n we have that the…

Let $k$ be a field, and let $L$ be an \'etale k-algebra of finite rank. If $a$ is a nonzero element in $k$, let $X_a$ be the affine variety defined by the norm equation $N_{L/k}(x) = a$. Assuming that $L$ has at least one factor that is a…

Given a number field $k$, a finitely generated subgroup $\mathcal{A}\subseteq k^\times$, and an integer $n\geq 3$, we study the distribution of $S_n$-extensions of $k$ such that the elements of $\mathcal{A}$ are norms. For $n\leq 5$, and…

For a field extension $L/K$ we consider maps that are quadratic over $L$ but whose polarisation is only bilinear over $K$. Our main result is that all such are automatically quadratic forms over $L$ in the usual sense if and only if $L/K$…

Fix a number field $K$. For each nonzero $\alpha \in \mathbb{Z}_K$, let $\nu(\alpha)$ denote the number of distinct, nonassociate irreducible divisors of $\alpha$. We show that $\nu(\alpha)$ is normally distributed with mean proportional to…

Let K be a Galois number field of prime degree $\ell$. Heilbronn showed that for a given $\ell$ there are only finitely many such fields that are norm-Euclidean. In the case of $\ell=2$ all such norm-Euclidean fields have been identified,…

Let $k$ be a field and $X \subset P^3_{k}$ a smooth cubic surface. Let $\Delta \subset Pic(X)$ be the finite index subgroup spanned by norms of lines on $X_{K}$ for $K$ running through the finite separable extensions of $k$. The quotient…