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A Theorem of Hou, Leung and Xiang generalised Kneser's addition Theorem to field extensions. This theorem was known to be valid only in separable extensions, and it was a conjecture of Hou that it should be valid for all extensions. We give an alternative proof of the theorem that also holds in the non-separable case, thus solving Hou's conjecture. This result is a consequence of a strengthening of Hou et al.'s theorem that is a transposition to extension fields of an addition theorem of Balandraud., 17 pages

Let k be a base field, K be a field containing k and L/K be a field extension of degree n. The essential dimension ed(L/K) over k is a numerical invariant measuring "the complexity" of L/K. Of particular interest is $\tau$(n) = max { ed(L/K) | L/K is a separable extension of degree n}, also known as the essential dimension of the symmetric group $S_n$. The exact value of $\tau$(n) is known only for n $\leq$ 7. In this paper we assume that k is a field of characteristic p > 0 and study the essential dimension of inseparable extensions L/K. Here the degree n = [L:K] is replaced by a pair (n, e) which accounts for the size of the separable and the purely inseparable parts of L/K respectively, and \tau(n) is replaced by $\tau$(n, e) = max { ed(L/K) | L/K is a field extension of type (n, e)}. The symmetric group $S_n$ is replaced by a certain group scheme $G_{n,e}$ over k. This group is neither finite nor smooth; nevertheless, computing its essential dimension turns out to be easier than computing the essential dimension of $S_n$. Our main result is a simple formula for \tau(n, e)., Comment: 18 pages

Let $A=\mathbb{F}_q[T]$ be the polynomial ring over $\mathbb{F}_q$, and $F$ be the field of fractions of $A$. Let $\phi$ be a Drinfeld $A$-module of rank $r\geq 2$ over $F$. For all but finitely many primes $\mathfrak{p}\lhd A$, one can reduce $\phi$ modulo $\mathfrak{p}$ to obtain a Drinfeld $A$-module $\phi\otimes\mathbb{F}_\mathfrak{p}$ of rank $r$ over $\mathbb{F}_\mathfrak{p}=A/\mathfrak{p}$. The endomorphism ring $\mathcal{E}_\mathfrak{p}=\mathrm{End}_{\mathbb{F}_\mathfrak{p}}(\phi\otimes\mathbb{F}_\mathfrak{p})$ is an order in an imaginary field extension $K$ of $F$ of degree $r$. Let $\mathcal{O}_\mathfrak{p}$ be the integral closure of $A$ in $K$, and let $\pi_\mathfrak{p}\in \mathcal{E}_\mathfrak{p}$ be the Frobenius endomorphism of $\phi\otimes\mathbb{F}_\mathfrak{p}$. Then we have the inclusion of orders $A[\pi_\mathfrak{p}]\subset \mathcal{E}_\mathfrak{p}\subset \mathcal{O}_\mathfrak{p}$ in $K$. We prove that if $\mathrm{End}_{F^\mathrm{alg}}(\phi)=A$, then for arbitrary non-zero ideals $\mathfrak{n}, \mathfrak{m}$ of $A$ there are infinitely many $\mathfrak{p}$ such that $\mathfrak{n}$ divides the index $\chi(\mathcal{E}_\mathfrak{p}/A[\pi_\mathfrak{p}])$ and $\mathfrak{m}$ divides the index $\chi(\mathcal{O}_\mathfrak{p}/\mathcal{E}_\mathfrak{p})$. We show that the index $\chi(\mathcal{E}_\mathfrak{p}/A[\pi_\mathfrak{p}])$ is related to a reciprocity law for the extensions of $F$ arising from the division points of $\phi$. In the rank $r=2$ case we describe an algorithm for computing the orders $A[\pi_\mathfrak{p}]\subset \mathcal{E}_\mathfrak{p}\subset \mathcal{O}_\mathfrak{p}$, and give some computational data., Comment: Journal of Number Theory (David Goss Memorial Issue), to appear

We extend the concept of the difference dimension polynomial of a difference field extension to difference local algebras. The main theorem of the paper establishes the existence and form of the dimension polynomial associated with the localization of a finitely generated well-mixed difference algebra at a prime reflexive difference ideal.

By a result of Kantor, any subgroup of GL(n, q) containing a Singer cycle normalizes a field extension subgroup. This result has as a consequence a projective analogue, and this paper gives the details of this deduction, showing that any subgroup of PΓL(n, q) containing a projective Singer cycle normalizes the image of a field extension subgroup GL(n/s, qs) under the canonical homomorphism GL(n, q) → PGL(n, q), for some divisor s of n, and so is contained in the image of ΓL(n/s, qs) under the canonical homomorphism ΓL(n, q) → PΓL(n, q). The actions of field extension subgroups on V (n, q) are also investigated. In particular, we prove that any field extension subgroup GL(n/s, qs) of GL(n, q) has a unique orbit on s-dimensional subspaces of V (n, q) of length coprime to q. This orbit is a Desarguesian s-partition of V (n, q).

Let F be a field of characteristic p > 0 . Let Ω n ( F ) be the F-vector space of n-differentials of F over F p . Let K = F ( g ) be the function field of an irreducible polynomial g in m ⩾ 1 variables over F. We derive an explicit description of the kernel of the restriction map Ω n ( F ) → Ω n ( K ) . As an application in the case p = 2 , we determine the kernel of the restriction map when passing from the Witt ring (resp. graded Witt ring) of symmetric bilinear forms over F to that over such a function field extension K.

The cube root Ramanujan formulas are explained from the point of view of Galois theory. Let F be a cyclic cubic extension of a field K. It is proved that the normal closure over K of a pure cubic extension of F contains a certain pure cubic extension of K. The proposed proof can be generalized to radicals of any prime degree q. In the case where the base field K is the field of rational numbers and the field F is embedded in the cyclotomic extension obtained by adding the pth roots of unity, the corresponding simple radical extension of the field of rational numbers is explicitly constructed. The proof of the main result illustrates Hilbert’s Theorem 90. An example of a particular formula generalizing Ramanujan’s formulas for degree 5 is given. A necessary condition for nested radical expressions of depth 2 to be contained in the normal closure of a pure cubic extension of the field F is given.

Let F be a field of characteristic 2 and let E / F be a field extension of degree 4. We determine the kernel W q ( E / F ) of the restriction map W q F → W q E between the Witt groups of nondegenerate quadratic forms over F and over E, completing earlier partial results by Ahmad, Baeza, Mammone and Moresi. We also deduce the corresponding result for the Witt kernel W ( E / F ) of the restriction map W F → W E between the Witt rings of nondegenerate symmetric bilinear forms over F and over E from earlier results by the first author. As an application, we describe the 2-torsion part of the Brauer kernel for such extensions.

Let k be a field, K/k be a quadratic separable field extension, and 𝒜 a finite dimensional central simple algebra over K. If k is global or the field of fractions of a two-dimensional excellent henselian local domain with an algebraically closed residue field of characteristic zero and the degree of 𝒜 is odd, we prove that all K/k-involutions on 𝒜 are cyclic.

Suppose E is an elliptic curve over $\Bbb Q$, and p>3 is a split multiplicative prime for E. Let q ≠ p be an auxiliary prime, and fix an integer m coprime to pq. We prove the generalised Mazur–Tate–Teitelbaum conjecture for E at the prime p, over number fields $K\subset \Bbb Q\big(\mu_{{q^{\infty}}},\;\!^{q^{\infty}\!\!\!\!}\sqrt{m}\big)$ such that p remains inert in $K\cap\Bbb Q(\mu_{{q^{\infty}}})^+$. The proof makes use of an improved p-adic L-function, which can be associated to the Rankin convolution of two Hilbert modular forms of unequal parallel weight.

In characteristic zero, local monomialization is true along any valuation. However, we have recently shown that local monomialization is not always true in positive characteristic, even in two dimensional algebraic function fields. In this paper we show that local monomialization is true for defectless extensions of two dimensional excellent local rings, extending an earlier result of Piltant and the author for two dimensional algebraic function fields over an algebraically closed field. We also give theorems showing that in many cases there are good stable forms of the extension of associated graded rings in a finite separable field extension.

Let k be an algebraically closed field of arbitrary characteristic, let K / k {K/k} be a finitely generated field extension and let X be a separated scheme of finite type over K. For each prime ℓ {\ell} , the absolute Galois group of K acts on the ℓ {\ell} -adic étale cohomology modules of X. We prove that this family of representations varying over ℓ {\ell} is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure of K of the kernels of the representations for all ℓ {\ell} become linearly disjoint over a finite extension of K. In doing this, we also prove a number of interesting facts on the images and on the ramification of this family of representations.

In previous work, Ohno conjectured, and Nakagawa proved, relations between the counting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of `extra functional equations' involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms. In the present paper we generalize their result by proving a similar identity relating certain degree l fields with Galois groups D_l and F_l respectively, for any odd prime l, and in particular we give another proof of the Ohno-Nakagawa relation without appealing to binary cubic forms., Version 2, 16 pages, to appear in Compositio

Let k be field of characteristic ≠ 2, 3. Assume that k contains more than 11 elements and let K be an algebraic field extension of k. The present paper describes subgroups of the general linear group GL 7 ( K ) that contain the Chevalley group G 2 ( k ) .

Let $K$ be a local field whose residue field has characteristic $p$ and let $L/K$ be a finite separable totally ramified extension of degree $n=ap^{\nu}$. The indices of inseparability $i_0,i_1,...,i_{\nu}$ of $L/K$ were defined by Fried in the case char$(K)=p$ and by Heiermann in the case char$(K)=0$; they give a refinement of the usual ramification data for $L/K$. The indices of inseparability can be used to construct "generalized Hasse-Herbrand functions" $\phi_{L/K}^j$ for $0\le j\le\nu$. In this paper we give an interpretation of the values $\phi_{L/K}^j(c)$ for natural numbers $c$. We use this interpretation to study the behavior of generalized Hasse-Herbrand functions in towers of field extensions., Comment: 16 pages

The Weil restriction functor, introduced in the late fifties, was recently extended by Karpenko to the category of Chow motives with integer coefficients. In this article we introduce the noncommutative (=NC) analogue of the Weil restriction functor, where schemes are replaced by dg algebras, and extend it to Kontsevich's categories of NC Chow motives and NC numerical motives. Instead of integer coefficients, we work more generally with coefficients in a binomial ring. Along the way, we extend Karpenko's functor to the classical category of numerical motives, and compare this extension with its NC analogue. As an application, we compute the (NC) Chow motive of the Weil restriction of every smooth projective scheme whose category of perfect complexes admits a full exceptional collection. Finally, in the case of central simple algebras, we describe explicitly the NC analogue of the Weil restriction functor using solely the degree of the field extension. This leads to a “categorification” of the classical corestriction homomorphism between Brauer groups.

Let K be a finite algebraic extension of the field of rational functions in one indeterminate over a finite field and let [Formula: see text] denote an algebraic closure of K. We count points in projective space [Formula: see text] with given height and generating a quadratic extension of K. If n > 2, we derive an asymptotic estimate for the number of such points as the height tends to infinity. Such estimates are analogous to previous results of Schmidt where the field K is replaced by the field of rational numbers ℚ.

In this paper, we present an overview of some of our research on the Levi-Civita fields R and C . R (resp. C ) is the smallest non-Archimedean field extension of the real (resp. complex) numbers that is Cauchy-complete and real closed (resp. algebraically closed); in fact, R is small enough to allow for the calculus on the field to be implemented on a computer and used in applications such as the fast and accurate computation of the derivatives of real functions as "differential quotients" up to very high orders. We summarize the convergence and analytical properties of power series, showing that they have the same smoothness behavior as real and complex power series; we present a Lebesgue-like measure and integration theory on the Levi-Civita field R ; we discuss solutions to one-dimensional and multi-dimensional optimization problems based on continuity and differentiability concepts that are stronger than the topological ones; and we give a brief summary of the results of our ongoing work on developing a non-Archimedean operator theory on a Banach space over C .

A general theorem on the purity of invariant field extensions is proved. Using it, a criterion of rational triangulability of connected solvable affine algebraic subgroups of the Cremona groups is obtained. This criterion is applied for proving the existence of rationally nontriangulable subgroups of the above form and for proving their stable rational triangulability. The latter property answers in the affirmative Bass’ Triangulability Problem in the stable range. A general construction of all rationally triangulable connected solvable affine algebraic subgroups of the Cremona groups is obtained. As an application, a classification of all rationally triangulable connected one-dimensional unipotent affine algebraic subgroups of the Cremona groups up to conjugacy is given.

In this chapter, we study automorphism groups of fields and introduce Galois groups of finite field extensions. The term “Galois group” is often reserved for automorphism groups of Galois field extensions, which we define and study in Chap. 9. The terminology used in this book is very common and has several advantages in textbooks (i.e. it is easier to formulate exercises). A central result of this chapter is Artin’s lemma, which is a key result in the modern presentation of Galois theory. In the exercises, we find Galois groups of many field extensions and we use also use this theorem for various problems on field extensions and their automorphism groups.

We begin this chapter with some basic facts about vector spaces. These will be familiar (at least in the case of real vector spaces) to those readers who have studied linear algebra. We then focus our attention on the particular case of a field extension. A number of properties of field extensions are discussed. Let F be a field and \(f(x)\in F[x]\) a nonconstant polynomial. We demonstrate how to create a field extension in which f(x) splits into a product of polynomials of degree 1. This leads to a classification of all finite fields.

We now apply our knowledge of field extensions in order to answer three questions posed by the ancient Greeks.

Let $\mathbb{K}$ be a discrete field and $(V, \phi)$ a flow over the category of locally linearly compact $\mathbb{K}$-spaces. Here we give the formulas to compute the topological entropy of $(V,\phi)$ subject to the extension or the restriction of scalars., Comment: conference paper

In this chapter, we look at three aspects of the theory of abstract fields: the discovery that all finite fields are ‘Galois’ fields; Dedekind’s presentation of the ‘Galois correspondence’ between groups and field extensions; and the emergence of the concept of an abstract field.

(I) The problem of “solving algebraic equations by radicals” (where by algebraic equations we mean those of the form f ( x ) = 0, for f a non-zero polynomial with rational coefficients) was one of the most important topics in mathematics from the earliest periods of antiquity until the 19th century. The Babylonians, and later the Greeks, already knew how to solve quadratic equations, although their formulas were more complex than those we use today, since their notation was inferior and they lacked the concepts of zero and negative numbers. Cubic equations were solved by S. del Ferro (around 1515) and N. Tartaglia (whose contributions were published in 1545 by G. Cardano and are often incorrectly attributed to the latter); quartic equations were solved by L. Ferrari using a method that was also published in Cardano’s Ars Magna together with the method proposed by Tartaglia. In 1576, R. Bombelli compiled a summary of all of this work using simpler notation in Algebra . However, all efforts to solve quintic equations were unsuccessful from the end of the 16th century until the early 19th century. J.-L. Lagrange was arguably the first to recognize the underlying reasons in his memoirs from 1770–1771. Finally, P. Ruffini attempted to show that the general quintic equation cannot be solved by radicals in a series of dense and controversial memoirs gradually published between 1799 and 1813. One proof that was entirely correct but weighed down by long calculations was provided by N. Abel in 1824; in 1826, he showed.

Let $k$ be a field, $n \geqslant 5$ be an integer, $x_1, \dots, x_n$ be independent variables and $L_n = k(x_1, \dots, x_n)$. The symmetric group $S_n$ acts on $L_n$ by permuting the variables, and the projective linear group ${\rm PGL}_2$ acts by applying (the same) fractional linear transformation to each varaible. The fixed field $L_n^{{\rm PGL}_2}$ is called "the field of cross-ratios". Let $S \subset S_n$ be a subgroup. The Noether Problem asks whether the field extension $L_n^S/k$ is rational, and the Noether Problem for cross-ratios asks whether $K_n^S/k$ is rational. In an effort to relate these two problems, H. Tsunogai posed the following question: Is $L_n^S$ rational over $K_n^S$? He answered this question in several situations, in particular, in the case where $S = S_n$. In this paper we extend his results by recasting the problem in terms of Galois cohomology. Our main theorem asserts that the following conditions on a subgroup $S \subset S_n$ are equivalent: (a) $L_n^S$ is rational over $K_n^S$, (b) $L_n^S$ is unirational over $K_n^S$, (c) $S$ has an orbit of odd order in $\{1, \dots, n \}$., Comment: 7 pages

The purpose of this note is to give a number of open problems on matching theory and their relation to the well-known results in this area. We also give a linear analogue of the acyclic matchings.

Let $A$ be a right noetherian algebra over a field $k$. If the base field extension $A \otimes_k K$ remains right noetherian for all extension fields $K$ of $k$, then $A$ is called stably right noetherian over $k$. We develop an inductive method to show that certain algebras of finite Gelfand-Kirillov dimension are stably noetherian, using critical composition series. We use this to characterize which algebras satisfying a polynomial identity are stably noetherian. The method also applies to many $\mathbb{N}$-graded rings of finite global dimension; in particular, we see that a noetherian Artin-Schelter regular algebra must be stably noetherian. In addition, we study more general variations of the stably noetherian property where the field extensions are restricted to those of a certain type, for instance purely transcendental extensions., Comment: 24 pages

The class numbers $h^{+}$ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows and methods employing Leopoldt's decomposition of the class number become hard to use when the field extension is not cyclic of prime power. This is why other methods have been developed which approach the problem from different angles. In this paper we extend one of these methods that was designed for real cyclotomic fields of prime conductor, and we make it applicable to real cyclotomic fields of conductor equal to the product of two distinct odd primes. The main advantage of this method is that it does not exclude the primes dividing the order of the Galois group, in contrast to other methods. We applied our algorithm to real cyclotomic fields of conductor $

We show that each real-representable matroid is a minor of a complex-representable excluded minor for real-representability. More generally, for an infinite field $\mathbb{F}_1$ and a field extension $\mathbb{F}_2$, if $\mathbb{F}_1$-representability is not equivalent to $\mathbb{F}_2$-representability, then each $\mathbb{F}_1$-representable matroid is a minor of a $\mathbb{F}_2$-representable excluded minor for $\mathbb{F}_1$-representability., Comment: 14 pages, 2 figures. Submitted for publication in JCTb

Categorical equivalences between block algebras of finite groups - such as Morita and derived equivalences - are well-known to induce character bijections which commute with the Galois groups of field extensions. This is the motivation for attempting to realise known Morita and derived equivalences over non splitting fields. This article presents various result on the theme of descent. We start with the observation that perfect isometries induced by a virtual Morita equivalence induce isomorphisms of centers in non-split situations, and explain connections with Navarro's generalisation of the Alperin-McKay conjecture. We show that Rouquier's splendid Rickard complex for blocks with cyclic defect groups descends to the non-split case. We also prove a descent theorem for Morita equivalences with endopermutation source., Theorem 1.13 has been improved

A Hopf Galois structure on a finite field extension $L/K$ is a pair $(\mathcal{H},��)$, where $\mathcal{H}$ is a finite cocommutative $K$-Hopf algebra and $��$ a Hopf action. In this paper, we present several results on Hopf Galois structures on odd prime power degree separable field extensions. We prove that if a separable field extension of odd prime power degree has a Hopf Galois structure of cyclic type, then it has no structure of noncyclic type. We determine the number of Hopf Galois structures of cyclic type on a separable field extension of degree $p^n$, $p$ an odd prime, such that the Galois group of its normal closure is a semidirect product $C_{p^n}\rtimes C_D$ of the cyclic group of order $p^n$ and a cyclic group of order $D$, with $D$ prime to $p$. We characterize the transitive groups of degree $p^3$ which are Galois groups of the normal closure of a separable field extension having some cyclic Hopf Galois structure and determine the number of those. We prove that if a separable field extension of degree $p^3$ has a nonabelian Hopf Galois structure then it has an abelian structure whose type has the same exponent as the nonabelian type. We obtain that, for $p>3$, the two abelian noncyclic Hopf Galois structures do not occur on the same separable extension of degree $p^3$. We present a table which gives the number of Hopf Galois structures of each possible type on a separable extension of degree $27$ to illustrate that for $p=3$, all four noncyclic Hopf Galois structures may occur on the same extension. Finally, putting together all previous results, we list all possible sets of Hopf Galois structure types on a separable extension of degree $p^3$, for $p>3$ a prime.

In this paper, we introduce the notions of matching matrices in groups and vector spaces, which lead to some necessary conditions for existence of acyclic matching in abelian groups and its linear analogue. We also study the linear local matching property in field extensions to find a dimension criterion for linear locally matchable bases. Moreover, we define the weakly locally matchable subspaces and we investigate their relations with matchable subspaces. We provide an upper bound for the dimension of primitive subspaces in a separable field extension. We employ MATLAB coding to investigate the existence of acyclic matchings in finite cyclic groups. Finally, a possible research problem on matchings in n-groups is presented. Our tools in this paper mix combinatorics and linear algebra.

A central conjecture in inverse Galois theory, proposed by D\`{e}bes and Deschamps, asserts that every finite split embedding problem over an arbitrary field can be regularly solved. We give an unconditional proof of a consequence of this conjecture, namely that such embedding problems can be regularly solved if one waives the requirement that the solution fields are normal. This extends previous results of M. Fried, Takahashi, Deschamps, and the last two authors concerning the realization of finite groups as automorphism groups of field extensions.

In this chapter, we show that equations solvable by radicals are characterized by the solvability of their Galois groups. This immediately implies that general equations of degree 5 and above are not solvable by radicals. If one has a more modest goal to prove that the fifth degree general equation over a number field is not solvable by radicals, then there exists a simple argument by Nagell which only requires limited knowledge of field extensions and no knowledge of Galois theory. We consider Nagell’s proof in the exercises. This chapter further outlines Weber’s theorem on irreducible equations of prime degree (at least 5) with only two nonreal zeros, which are examples of non-solvable equations. We further discuss Galois’ classical theorem, which gives a characterization of irreducible solvable polynomials of prime degree. Both Galois’ and Weber’s results give examples of concrete unsolvable polynomials over the rational numbers. The solvability by real radicals in connection with “casus irreducibilis” is also discussed.

Using Semaev's summation polynomials, we derive a new equation for the $\mathbb{F}_q$-rational points of the trace zero variety of an elliptic curve defined over $\mathbb{F}_q$. Using this equation, we produce an optimal-size representation for such points. Our representation is compatible with scalar multiplication. We give a point compression algorithm to compute the representation and a decompression algorithm to recover the original point (up to some small ambiguity). The algorithms are efficient for trace zero varieties coming from small degree extension fields. We give explicit equations and discuss in detail the practically relevant cases of cubic and quintic field extensions., Comment: 23 pages, to appear in Designs, Codes and Cryptography

Let A be an abelian variety defined over a number field F. For a prime number $\ell$, we consider the field extension of F generated by the $\ell$-powered torsion points of A. According to a conjecture made by Rasmussen and Tamagawa, if we require these fields to be both a pro-$\ell$ extension of $F(\mu_{\ell^{\infty}})$ and unramified away from $\ell$, examples are quite rare. Indeed, it is expected that for a fixed dimension and field of definition, there exists such an abelian variety for only a finite number of primes. We prove a uniform version of the conjecture in the case where the abelian varieties are elliptic curves with complex multiplication. In addition, we provide explicit bounds in cases where the number field has degree less than or equal to 100.

We construct full-diversity, arbitrary rate STBCs for specific number of transmit antennas over an apriori specified signal set using twisted Laurent series rings. Constructing full-diversity space-time block codes from algebraic constructions like division algebras has been done by Shashidhar et al. Constructing STBCs from crossed product algebras arises this question in mind that besides these constructions, which one of the well-known division algebras are appropriate for constructing space-time block codes. This paper deals with twisted Laurent series rings and their subrings twisted function fields, to construct STBCs. First, we introduce twisted Laurent series rings over field extensions of $\mathbb{Q}$. Then, we generalize this construction to the case that coefficients come from a division algebra. Finally, we use an algorithm to construct twisted function fields, which are noncrossed product division algebras, and we propose a method for constructing STBC from them.

We classify all invariants of the functor $I^n$ (powers of the fundamental ideal of the Witt ring) with values in $A$, that it to say functions $I^n(K)\rightarrow A(K)$ compatible with field extensions, in the cases where $A(K)=W(K)$ is the Witt ring and $A(K)=H^*(K,\mu_2)$ is mod 2 Galois cohomology. This is done in terms of some invariants $f_n^d$ that behave like divided powers with respect to sums of Pfister forms, and we show that any invariant of $I^n$ can be written uniquely as a (possibly infinite) combination of those $f_n^d$. This in particular allows to lift operations defined on mod 2 Milnor K-theory (or equivalently mod 2 Galois cohomology) to the level of $I^n$. We also study various properties of these invariants, including behaviour under products, similitudes, residues for discrete valuations, and restriction from $I^n$ to $I^{n+1}$. The goal is to use this to study invariants of algebras with involutions in future articles.

Galois Theory, a wonderful part of mathematics with historical roots date back to the solution of cubic and quantic equations in the sixteenth century. However, beside understanding the roots of polynomials, Galois Theory also gave birth to many of the central concepts of modern algebra, including groups and fields. In particular, this theory is further great due to primarily for two factors: first, its surprising link between the group theory and the roots of polynomials and second,the elegance of its presentation. This theory is often descried as one of the most beautiful parts of mathematics. Here I have specially worked on field extensions. To understand the basic concept behind fundamental theory, some necessary Theorems, Lammas and Corollaries are added with suitable examples containing Lattice Diagrams and Tables. In principle, I have presented and solved a number of complex algebraic problems with the help of Galois theory which are designed in the context of various rational and complex numbers.

In this paper we introduce the local Nori fundamental group scheme of a reduced scheme or algebraic stack over a perfect field $k$. We give particular attention to the case of fields: to any field extension $K/k$ we attach a pro-local group scheme over $k$. We show how this group has many analogies, but also some crucial differences, with the absolute Galois group. We propose two conjectures, analogous to the classical Neukirch-Uchida Theorem and Abhyankar Conjecture, providing some evidence in their favor. Finally we show that the local fundamental group of a normal variety is a quotient of the local fundamental group of an open, of its generic point (as it happens for the \'etale fundamental group) and even of any smooth neighborhood., Comment: 17 pages

Let $k$ be a field, with absolute Galois group $\Gamma$. Let $A/k$ be a finite \'etale group scheme of multiplicative type, i.e. a discrete $\Gamma$-module. Let $n \geq 2$ be an integer, and let $x \in H^n(k,A)$ be a cohomology class. We show that there exists a countable set $I$, and a familiy $(X_i)_{i \in I}$ of (smooth, geometrically integral) $k$-varieties, such that the following holds. For any field extension $l/k$, the restriction of $x$ vanishes in $H^n(l,A)$ if and only if (at least) one of the $X_i$'s has an $l$-point. We moreover show that the $X_i$'s can be made into an ind-variety. In the case $n=2$, we note that one variety is enough., Comment: 13 pages

We give a sharp divisibility bound, in terms of g, for the degree of the field extension required to realize the endomorphisms of an abelian variety of dimension g over an arbitrary number field; this refines a result of Silverberg. This follows from a stronger result giving the same bound for the order of the component group of the Sato–Tate group of the abelian variety, which had been proved for abelian surfaces by Fite–Kedlaya–Rotger–Sutherland. The proof uses Minkowski’s reduction method, but with some care required in the extremal cases when p equals 2 or a Fermat prime.

We are interested in characterising pairs $S,T$ of $F$-linear subspaces in a field extension $L/F$ such that the linear span $ST$ of the set of products of elements of $S$ and of elements of $T$ has small dimension. Our central result is a linear analogue of Vosper's Theorem, which gives the structure of vector spaces $S, T$ in a prime extension $L$ of a finite field $F$ for which $\dim_FST =\dim_F S+\dim_F T-1,$ when $\dim_F S, \dim_F T\ge 2$ and $\dim_F ST\le [L:F]-2$., 33 pages

We establish an analytic Hasse principle for linear spaces of affine dimension $m$ on a complete intersection over an algebraic field extension $\mathbb{K}$ of $\mathbb{Q}$ . The number of variables required to do this is no larger than what is known for the analogous problem over $\mathbb{Q}$ . As an application, we show that any smooth hypersurface over $\mathbb{K}$ whose dimension is large enough in terms of the degree is $\mathbb{K}$ -unirational, provided that either the degree is odd or $\mathbb{K}$ is totally imaginary.