Narrow Results

Let $f(x)=x^8+a{x}^4+b$ be an irreducible polynomial with rational coefficients and Galois group $\mathrm{\text{Gal}}(f)$. We extend previous results to give an elementary classification of $\mathrm{\text{Gal}}(f)$, identified as a transitive subgroup of $S_8$ up to conjugacy. We show $\mathrm{\text{Gal}}(f)$ is one of 12 possibilities and can be determined by considering the squareness of at most 11 rational numbers; each number is an expression involving $a$ and $b$. We give several applications of our results.

Let $F$ be a field of characteristic different from $2$, and let $L=F(\sqrt{b+2c\sqrt{a}})$ be a tower of two quadratic field extensions. We construct a filtration on the relative Brauer group $\mathrm{\text{Br}}(L/F)$ and compute its factors in terms of certain norm groups. The answer depends on whether the extension $L/F$ is biquadratic or not. As a consequence we obtain a description of the group ${F^{\ast }}^2{N}_{E/F}{E}^{\ast }$ for a Galois field extension $E/F$ with dihedral Galois group. Also, we prove that for any $2$-torsion abelian group $G$ and any biquadratic extension $L/F$ there exists a field extension $K/F$ linearly disjoint with $L/F$ such that $\mathrm{\text{Br}}(LK/K){/}_2\mathrm{\text{Br}}(LK/K)\simeq G$.

In this paper we present applications of polynomial algebra for the problem of computing Nash equilibria of a subclass of finite normal form games.We characterize Nash equilibria of a normal form game as solutions to a system of polynomial equations and define the subclass of games under consideration. We present an algebraic method for deciding membership decision to the subclass of games. A method based on group action to compute all Nash equilibria of the subclass of games is presented with examples to show working of the methods. We also present some related results and discuss properties of the subclass of games.

Given a rational matrix A, and a set of rational matrices B, C,… which commute with A, we give polynomial time algorithms to compute exactly the Jordan Normal Form of A, as well as the transformed matrices of B, C,…. We also obtain the transformation matrix and its inverse exactly in polynomial time.

Suppose $K$ is a finite field extension of ${\mathbb{Q}}_p$ containing a primitive $p$th root of unity. Let $K_{<p}$ be a maximal $p$-extension of $K$ with the Galois group of period $p$ and nilpotent class $<p$. In this paper, we develop formalism which allows us to study the structure of ${\mathrm{\Gamma }}_{<p}=\mathrm{\text{Gal}}(K_{<p}/K)$ via methods of Lie theory. In particular, we introduce an explicit construction of a Lie ${𝔽}_p$-algebra $L$ and an identification ${\mathrm{\Gamma }}_{<p}=G(L)$, where $G(L)$ is a $p$-group obtained from the elements of $L$ via the Campbell–Hausdorff composition law. In the next paper, we apply this formalism to describe the ramification filtration ${\{{\mathrm{\Gamma }}_{<p}^{(v)}\}}_{v\ge 0}$ and an explicit form of the Demushkin relation for ${\mathrm{\Gamma }}_{<p}$.

Suppose $K$ is a finite field extension of ${\mathbb{Q}}_p$ containing a primitive $p$th root of unity. Let ${\mathrm{\Gamma }}_{<p}$ be the maximal quotient of period $p$ and nilpotent class $<p$ of the Galois group of a maximal $p$-extension of $K$. We describe the ramification filtration ${\{{\mathrm{\Gamma }}_{<p}^{(v)}\}}_{v\ge 0}$ and relate it to an explicit form of the Demushkin relation for ${\mathrm{\Gamma }}_{<p}$. The results are given in terms of Lie algebras attached to the appropriate $p$-groups by the classical equivalence of the categories of $p$-groups and Lie algebras of nilpotent class $<p$.

Let $n\in \mathbb{Z}$ with $n\ge 3$. Let $S_n$ and $A_n$ denote, respectively, the symmetric group and alternating group on $n$ letters. Let $m$ be an indeterminate, and define

For each integer $n\ge 2$, we identify new infinite families of monogenic trinomials $f(x)=x^n+A{x}^m+B$ with non-squarefree discriminant, many of which have small Galois group. Moreover, in certain situations when $A=B\ge 2$ with fixed $n$ and $m$, we produce asymptotics on the number of such trinomials with $A\le X$.

In this paper we significantly extend the limits of existing algebraic algorithms for computing Lagrange resolvents. Such algorithms are fundamental in effective Galois theory. However, they usually are of high complexity. For the case of absolute resolvents, N. Rennert has shown the value of a multimodular approach to reduce the complexity in space and time. We improve and generalize his work to the computation of any resolvent. In addition, we introduce a decomposition formula which allows us to split modular resolvents into resolvents of smaller degrees, thus both speeding computations and making it possible to efficiently parallelize them.

In this paper, we study which field F can be represented as sums of two proper subfields. We prove that a field F is a sum of two proper subfields if and only if the transcendence degree of F over its prime subfield is infinite. This answers a question posed in the paper [M. Kȩpczyk and R. Mazurek, On the representation of fields as finite sums of proper subfields, Results Math.75(2) (2020) 65].

We prove the existence and finiteness theorems for integral representations stable under Galois operation. An explicit construction of the realization fields for representations of finite groups stable under the natural operation of the Galois group is given. We also compare the representations over fields and the rings of integers, and give a quantitative result on the rarity of integral Galois stable representations. There is a series of related conjectures and applications to arithmetic algebraic geometry, finite flat group schemes, positive definite quadratic lattices and Galois cohomology.

We characterize irreducible trinomials x⁶ + Ax + B with coefficients in a number field K which have Galois group C₆, S₃ or C₃ × S₃. This characterization relates these trinomials to the K-rational points on a genus 2 curve. We determine these trinomials explicitly in the case K = ℚ.

Let F be a field of prime characteristic p containing 𝔽_pⁿ as a subfield. We refer to q(X) = X^pn - X - a ∈ F[X] as a generalized Artin–Schreier polynomial. Suppose that q(X) is irreducible and let C_q(X) be the companion matrix of q(X). Then ad C_q(X) has such highly unusual properties that any A ∈ 𝔤𝔩(m) such that ad A has like properties is shown to be similar to the companion matrix of an irreducible generalized Artin–Schreier polynomial. We discuss close connections with the decomposition problem of the tensor product of indecomposable modules for a one-dimensional Lie algebra over a field of characteristic p, the problem of finding an explicit primitive element for every intermediate field of the Galois extension associated to an irreducible generalized Artin–Schreier polynomial, and the problem of finding necessary and sufficient conditions for the irreducibility of a family of polynomials.

Let p be a prime number and (F, v) a valued field. In this paper, we find a presentation for the p-torsion part of the Brauer group Br(F), by means of the valuation v. We only assume that F has a primitive pth root of the unity and the residue class field has characteristic not equal to p. This result naturally leads to consider valued fields that we call pre-p-henselian fields. It concerns valuations compatible with R_p, the p-radical of the field. To be precise, R_p is the radical of the skew-symmetric pairing which associates to a pair (a, b) the class of the symbol algebra (F; a, b) in BrF. In our main result, we state that pre-p-henselian fields are precisely the fields for which the Galois group of the maximal Galois p-extension admits a particular decomposition as a free pro-p product.

Let p₁ ≡ p₂ ≡ -q ≡ 1 (mod 4) be primes such that and . Put and d = p₁p₂q, then the bicyclic biquadratic field has an elementary Abelian 2-class group of rank 3. In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-Abelian Galois group of the second Hilbert 2-class field of 𝕂, we study the 2-class field tower of 𝕂, and we study the capitulation problem of the 2-classes of 𝕂 in its fourteen abelian unramified extensions of relative degrees two and four.

An associative ring with unity is called clean if each of its elements is the sum of an idempotent and a unit. A clean ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In a recent paper, Huang, Li and Yuan provided a complete characterization that when a group ring ${𝔽}_q{C}_{p^k}$ is ∗-clean, where ${𝔽}_q$ is a finite field and $C_{p^k}$ is a cyclic group of an odd prime power order $p^k$. They also provided a necessary condition and a few sufficient conditions for ${𝔽}_q{C}_n$ to be ∗-clean, where $C_n$ is a cyclic group of order $n$. In this paper, we extend the above result of Huang, Li and Yuan from ${𝔽}_q{C}_{p^k}$ to $𝔽G$ and provide a characterization of ∗-clean group rings $𝔽G$, where $G$ is a finite abelian group and $𝔽$ is a field with characteristic not dividing the exponent of $G$.

We introduce a certified numerical algorithm to calculate Lagrange’s resolvent. This algebraic certification uses Galoisian ideal.

Let $f(x)=x^8+a{x}^4+b$ be an irreducible polynomial with rational coefficients, $K/Q$ the number field defined by $f$, and $G$ the Galois group of $f$. Let $g(x)=x^4+a{x}^2+b$, and let $G_4$ be the Galois group of $g$. We investigate the extent to which knowledge of the conjugacy class of $G_4$ in $S_4$ determines the conjugacy class of $G$ in $S_8$. We show that, in general, knowledge of $G_4$ does not automatically determine $G$, except when $G_4$ is isomorphic to $C_4$ (the cyclic group of order 4). In this case, we show $G$ is isomorphic to a non-split extension of $D_4$ (the dihedral group of order 8) by $C_4$. We also show that $G$ is completely determined when $G_4$ is isomorphic to $D_4$ and $4b-a^2$ is a perfect square. In this case, $G\simeq C_4\wr C_2\simeq (C_4\times C_4)⋊C_2$.

Let $C_n$ denote the cyclic group of order $n$, and let Hol($C_n$) denote the holomorph of $C_n$. In this paper, for any odd integer $m\ge 3$, we find necessary and sufficient conditions on an integer $A$, with $\left|A\right|\ge 3$, such that ${{ℱ}}_{m,A}(x)=x^{2m}+A{x}^m+1$ is irreducible over $\mathbb{Q}$. When $m=q\ge 3$ is prime and ${{ℱ}}_{q,A}(x)$ is irreducible, we show that the Galois group over $\mathbb{Q}$ of ${{ℱ}}_{q,A}(x)$ is isomorphic to either Hol($C_q$) or Hol($C_{2q}$), depending on whether there exists $y\in \mathbb{Z}$ such that $A^2-4=q{y}^2$. Finally, we prove that there exist infinitely many positive integers $A$ such that ${{ℱ}}_{q,A}(x)$ is irreducible over $\mathbb{Q}$ and that $\{1,𝜃,{𝜃}^2,\dots ,{𝜃}^{2q-1}\}$ is a basis for the ring of integers of $K=\mathbb{Q}(𝜃)$, where ${{ℱ}}_{q,A}(𝜃)=0$.

Let C be a curve, and l, l₀ be lines in the projective three space ℙ³. Consider a projection π_l: ℙ³ ⋯ → l₀ with center l, where l ∩ l₀ = ∅. Restricting π_l to C, we get a morphism π_l|_C: C → l₀ and an extension of fields (π_l|_C)* : k(l₀) ↪ k(C). We study the algebraic structure of the extension and the geometric structure of C. In particular, we study the structure of the Galois group and the number of Galois lines for some special cases.