Related papers: Galois extensions and subspaces of bilinear forms …

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 $\sym_K(L)$…

Let $L/K$ be a cyclic extension of degree $n = 2m$. It is known that the space $\text{Alt}_K(L)$ of alternating $K$-bilinear forms (skew-forms) on $L$ decomposes into a direct sum of $K$-subspaces $A^{\sigma^i}$ indexed by the elements of…

Let V be a vector space of dimension n over a field K and let Symm(V) denote the space of symmetric bilinear forms defined on V x V. Let M be a subspace of Symm(V). We investigate a variety of hypotheses concerning the rank of elements in M…

In \cite{GQ2008} R. Gow and R. Quinlan have cast a new look on the endomorphism algebra of a $K$-vector space $V$ of dimension $n$ assuming that $K$ has a Galois extension $L$ of degree $n$. In this approach the $K$-space $L$ may serve as a…

This note presents Galois theory for finite fields. It was written as a handout for the MAT401 course ``Polynomial equations and fields'' taught at the University of Toronto in Spring 2026. We use without proofs some basic properties of…

Let K be a field of characteristic different from 2 and let V be a vector space of dimension n over K. Let M be a non-zero subspace of symmetric bilinear forms defined on V x V and let r=rank(M) denote the set of different positive integers…

We introduce and study a class of field extensions that we call pre-Galois; viz. extensions that become Galois after some linearly disjoint Galois base change. Among them are geometrically Galois extensions of k(T), with k a field:…

Let k be a number field, and denote by k^[d] the compositum of all degree d extensions of k in a fixed algebraic closure. We first consider the question of whether all algebraic extensions of k of degree less than d lie in k^[d]. We show…

We study toric varieties over a field k that split in a Galois extension K/k using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation…

Let $p$ be prime, and $n,m \in \mathbb{N}$. When $K/F$ is a cyclic extension of degree $p^n$, we determine the $\mathbb{Z}/p^m\mathbb{Z}[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times p^m}$. With at most one exception, each…

We prove that the category of preordered groups contains two full reflective subcategories that give rise to some interesting Galois theories. The first one is the category of the so-called commutative objects, which are precisely the…

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…

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…

We studied framed deformations of two dimensional Galois representation of which the residue representation restrict to decomposition groups are scalars, and established a modular lifting theorem for certain cases. We then proved a family…

For a Galois extension $K/F$ with $\text{char}(K)\neq 2$ and $\text{Gal}(K/F) \simeq \mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$, we determine the $\mathbb{F}_2[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times 2}$. Although…

We classify all division algebras that are principal Albert isotopes of a cyclic Galois field extension of degree $n>2$ up to isomorphisms. We achieve a ``tight'' classification when the cyclic Galois field extension is cubic. The…

Let q be a power of a prime and let V be a vector space of finite dimension n over the field of order q. Let Bil(V) denote the set of all bilinear forms defined on V x V, let Symm(V) denote the subspace of Bil(V) consisting of symmetric…

We use Galois descent to construct central extensions of twisted forms of split simple Lie algebras over rings. These types of algebras arise naturally in the construction of Extended Affine Lie Algebras. The construction also gives…

Let $p$ be an irregular prime. Let $K=\Q(\zeta)$ be the $p$-cyclotomic field. From Kummer and class field theory, there exist Galois extensions $S/\Q$ of degree $p(p-1)$ such that $S/K$ is a cyclic unramified extension of degree $[S:K]=p$.…

In [2], an exhaustive construction is achieved for the class of all 4-dimensional unital division algebras over finite fields of odd order, whose left nucleus is not minimal and whose automorphism group contains Klein's four-group. We…