We study the common splitting fields of symbol algebras of degree $p^m$ over fields $F$ of $\operatorname{char}(F)=p$. We first show that if any finite number of such algebras share a degree $p^m$ simple purely inseparable splitting field, then they share a cyclic splitting field of the same degree. As a consequence, we conclude that every finite number of symbol algebras of degrees $p^{m_0},\dots,p^{m_t}$ share a cyclic splitting field of degree $p^{m_0+\dots+m_t}$. This generalization recovers the known fact that every tensor product of symbol algebras is a symbol algebra. We apply a result of Tignol's to bound the symbol length of classes in $\operatorname{Br}_{p^m}(F)$ whose symbol length when embedded into $\operatorname{Br}_{p^{m+1}}(F)$ is 2 for $p\in \{2,3\}$. We also study similar situations in other Kato-Milne cohomology groups, where the necessary norm conditions for splitting exist.
We use constructions of versal cohomology classes based on a new notion of "presentable functors," to describe a relationship between the problems of bounding symbol length in cohomology and of finding the minimal degree of a splitting field. The constructions involved are then also used to describe generic splitting varieties for degree 2 cohomology with coefficients in a commutative algebraic group of multiplicative type.
Let S be a symbol algebra. The trace form of S is computed and it is shown how this form can be used to determine whether S is a division algebra or not. In addition, the exterior powers of the trace form of S are computed.
Let A be a central simple algebra over a field F. Let k_1,\ldots, k_r be cyclic extensions of F such that k_1\otimes_F\cdots \otimes_F k_r is a field. We investigate conditions under which A is a tensor product of symbol algebras where each k_i is in a symbol F-algebra factor of the same degree as k_i. As an application, we give an example of an indecomposable algebra of degree 8 and exponent 2 over a field of 2-cohomological dimension 4.
This paper is concerned with the problem of determining the number of division algebras which share the same collection of finite splitting fields. As a corollary we are able to determine when two central division algebras may be distinguished by their finite splitting fields over certain fields.
We prove that if the maximal dimension of an anisotropic homogeneous polynomial form of prime degree $p$ over a field $F$ with $\operatorname{char}(F)=p$ is a finite integer $d$ greater than 1 then the symbol length of $p$-algebras of exponent $p$ over $F$ is bounded from above by $\left \lceil \frac{d-1}{p} \right \rceil-1$, and show that every two tensor products of symbol algebras of lengths $k$ and $\ell$ with $(k+\ell) p \geq d-1$ can be modified so that they share a common slot. For $p=2$, we obtain an upper bound of $\frac{u(F)}{2}-1$ for the symbol length, which is sharp when $I_q^3 F=…
In this paper, we will study the Möbius polynomial, an invariant of ranked posets that arises in the study of splitting algebras. We will present a formula for the Möbius polynomial of the direct product of posets in terms of the Möbius polynomials of the factors. We will then use this formula to calculate Hilbert series and graded trace generating functions associated to the splitting algebras of the Boolean algebra and the poset of factors of a natural number n.
Given a field $F$ of characteristic $3$ and division symbol $p$-algebras $[α,β)_{3,F}$ and $[α,γ)_{3,F}$ of degree $3$ over $F$, we prove that if $α\text{dlog}(β)\wedge \text{dlog}(γ)$ is trivial in the Kato-Milne cohomology group $H_3^3(F)$ then the algebras share a common splitting field which is an inseparable degree 3 extension of either $F$ or a quadratic extension of $F$. In the special case of quadratically closed fields, if $α\text{dlog}(β)\wedge \text{dlog}(γ)=0$, then they share an inseparable degree 3 extension of $F$.
Gelfand, Retakh, Serconek and Wilson, in \cite{GRSW}, defined a graded algebra $A_Γ$ attached to any finite ranked poset $Γ$ - a generalization of the universal algebra of pseudo-roots of noncommutative polynomials. This algebra has since come to be known as the splitting algebra of $Γ$. The splitting algebra has a secondary filtration related to the rank function on the poset and the associated graded algebra is denoted here by $A'_Γ$. We calculate the cohomology algebra (and coalgebra) of $A'_Γ$ explicitly. As a corollary to this calculation we have a proof that $A'_Γ$ is Koszul (respectivel…
In this paper we define pre-Malcev algebras and alternative quadri-algebras and prove that they generalize pre-Lie algebras and quadri-algebras respectively to the alternative setting. Constructions in terms bimodules, splitting of operations, and Rota-Baxter operators are discussed.
Splitting of operations for Hom-diassociative and Hom-triassociative Algebras
University of Carthage（迦太基大学）；Higher Institute of Sciences and Technology of Environment of Borj Cedria (ISTEUB)（布尔吉塞里亚环境科学与技术高等学院）；Université de Gafsa（加夫萨大学）；Facultté des sciences de Gafsa（加夫萨科学学院）；Département de Mathématiques, Faculté des Sciences, Université de Sfax（萨法克斯大学理学院数学系）；IRIMAS - Département de Mathématiques, Faculté des Sciences, Université de Haute Alsace（上阿尔萨斯大学理学院数学系 IRIMAS）
We bound the symbol length of elements in the Brauer group of a field $K$ containing a $C_m$ field (for example any field containing an algebraically closed field or a finite field), and solve the local exponent-index problem for a $C_m$ field $F$. In particular, for a $C_m$ field $F$, we show that every $F$ central simple algebra of exponent $p^t$ is similar to the tensor product of at most $len(p^t,F)\leq t(p^{m-1}-1)$ symbol algebras of degree $p^t$. We then use this bound on the symbol length to show that the index of such algebras is bounded by $(p^t)^{(p^{m-1}-1)}$, which in turn gives a…
It is well known that central simple algebras are split by suitable finite Galois extensions of their centers. A counterpart of this result was studied by Juan and Magid in the set up of differential matrix algebras, wherein Picard-Vessiot extensions that split matrix differential algebras were constructed. In this article, we exhibit instances of differential matrix algebras which are split by finite extensions. In some cases, we relate the existence of finite splitting extensions of a differential matrix algebra to the triviality of its tensor powers, and show in these cases, that orders of …
We prove that the maximal dimension of a $p$-central subspace of the generic symbol $p$-algebra of prime degree $p$ is $p+1$. We do it by proving the following number theoretic fact: let $\{s_1,\dots,s_{p+1}\}$ be $p+1$ distinct nonzero elements in the additive group $G=(\mathbb{Z}/p \mathbb{Z}) \times (\mathbb{Z}/p \mathbb{Z})$; then every nonzero element $g \in G$ can be expressed as $d_1 s_1+\dots+d_{p+1} s_{p+1}$ for some non-negative integers $d_1,\dots,d_{p+1}$ with $d_1+\dots+d_{p+1} \leq p-1$.
Given a field $F$, an étale extension $L/F$ and an Azumaya algebra $A/L$, one knows that there are extensions $E/F$ such that $A \otimes_F E$ is a split algebra over $L \otimes_F E$. In this paper we bound the degree of a minimal splitting field of this type from above and show that our bound is sharp in certain situations, even in the case where $L/F$ is a split extension. This gives in particular a number of generalizations of the classical fact that when the tensor product of two quaternion algebras is not a division algebra, the two quaternion algebras must share a common quadratic splitti…
In this paper we provide an algorithm to compute the product between two elements in a symbol algebra of degree n and we find an octonion non-division algebra in a symbol algebra of degree three. Starting from this last idea, we try to find an answer to the question if there are division symbol algebras of degree three. The answer is positive and we provide, using MAGMA software, some examples of division symbol algebras of degree 3 and of degree 5. Moreover, we will give some interesting applications of the symbol algebras in number theory.
In this article the $p$-essential dimension of generic symbols over fields of characteristic $p$ is studied. In particular, the $p$-essential dimension of the length $\ell$ generic $p$-symbol of degree $n+1$ is bounded below by $n+\ell$ when the base field is algebraically closed of characteristic $p$. The proof uses new techniques for working with residues in Milne-Kato $p$-cohomology and builds on work of Babic and Chernousov in the Witt group in characteristic 2. Two corollaries on $p$-symbol algebras (i.e, degree 2 symbols) result from this work. The generic $p$-symbol algebra of length $\…
In this paper we will study some properties of the matrix representations of symbol algebras of degree three, we study some equations with coefficients in these algebras, we find an octonion algebra in a symbol algebra of degree three, we define the Fibonacci symbol elements and we give some properties of them.
In this paper, we obtain some properties of the symbol algebras, starting from their connections with the quaternion and cyclic algebras over a field $K_{p},$% where $K$ is an algebraic number field, $p$ is a prime in $K$ and $K_{p}$ is the completion of $K$ with respect to $p-$ adic valuation, in the case when $K=\Bbb{Q}(\varepsilon), \varepsilon ^{3}=1,\varepsilon \neq 1.$
Let $K$ be a field with characteristic different from 2 and let $S$ be a symbol algebra over $K$. We compute the symmetric powers of hyperbolic quadratic forms over $K$. Also, we compute the symmetric powers of the quadratic trace form of $S$. In both cases we apply a generalised form of the Vandermonde convolution in the course of the computations.
A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper provides an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p-algebras) over global fields are crossed products. A closely related statement holds for division algebras over Henselian valued fields with global residue field. The existence of absolute Galois splitting fields in central simple algebras…
Let $p$ be an odd prime, let $K=\mathbb{Q}(ε)$ where $ε$ is a primitive cubic root of unity, and let $L$ be the Kummer field $\mathbb{Q}\left(ε, \sqrt[3]α\right)$. In this paper we obtain a characterization of the splitting behavior of the symbol algebras $\left( \frac{α,p}{K,ε}\right)$ and $\left( \frac{α,p^{h_{p}}}{K,ε}\right)$, where $h_{p}$ is the order in the class group $Cl\left(L\right)$ of a prime ideal of $\mathcal{O}_L$ which divides $p\mathcal{O}_L.$
We associate to a good cell decomposition of a manifold M a quadratic algebra and show that the Koszulity of the algebra implies a restriction on the Euler characteristic of M. For a two-dimensional manifold M the algebra is Koszul if and only if the Euler characteristic of M is two.
We study a finite dimensional quadratic graded algebra R defined from a finite ranked poset. This algebra has been central to the study of the splitting algebra of the poset, A, as introduced by Gelfand, Retakh, Serconek and Wilson . The algebra A is known to be quadratic when the poset satisfies a combinatorial condition known as uniform, and R is the quadratic dual of an associated graded algebra of A. We prove that R is Koszul and the poset is uniform if and only if the poset is Cohen-Macaulay. Koszulity of R implies Koszulity of A. We also show that when R is Koszul, the cohomology of the …
We propose an algorithm for finding zero divisors in quaternion algebras over quadratic number fields, or equivalently, solving homogeneous quadratic equations in three variables over $\mathbb{Q}(\sqrt{d})$ where $d$ is a square-free integer. The algorithm is randomized and runs in polynomial time if one is allowed to call oracles for factoring integers.
Let $A$ be a pseudocompact (or profinite) algebra, so $A=C^*$ where $C$ is a coalgebra. We show that the if the semiartinian part (the "Dickson" part) of every $A$-module $M$ splits off in $M$, then $A$ is semiartinian, also giving a positive answer in the case of algebras arising as dual of coalgebras (pseudocompact algebras), to a well known conjecture of Faith.
