On the non-existence of cyclic splitting fields for division algebras

2018 ◽  
Vol 30 (2) ◽  
pp. 385-395
Author(s):  
Mehran Motiee
Keyword(s):  
Division Algebra ◽  
Commutator Subgroup ◽  
Cyclic Extension ◽  
Roots Of Unity ◽  
Division Algebras ◽  
Group Of Units ◽  
Positive Integers ◽  
Special Classes

AbstractLetDbe a division algebra over its centerFof degreen. Consider the group{\mu_{Z}(D)=\mu_{n}(F)/Z(D^{\prime})}, where{\mu_{n}(F)}is the group of all then-th roots of unity in{F^{*}}, and{Z(D^{\prime})}is the center of the commutator subgroup of the group of units{D^{*}}ofD. It is shown that if{\mu_{Z}(D\otimes_{F}L)\neq 1}for someLcontaining all the primitive{n^{k}}-th roots of unity for all positive integersk, thenDis not split by any cyclic extension ofF. This criterion is employed to prove that some special classes of division algebras are not cyclically split.

Division Algebras of Prime Degree and Maximal Galois p-Extensions

2007 ◽  
Vol 59 (3) ◽  
pp. 658-672
Author(s):  
J. Mináč ◽  
A. Wadsworth
Keyword(s):  
Prime Number ◽  
Division Algebra ◽  
Roots Of Unity ◽  
Division Algebras ◽  
Prime Degree ◽  
Cyclic Algebra ◽  
Residue Fields ◽  
Residue Characteristic

AbstractLet p be an odd prime number, and let F be a field of characteristic not p and not containing the group μp of p-th roots of unity. We consider cyclic p-algebras over F by descent from L = F(μp). We generalize a theorem of Albert by showing that if μpn ⊆ L, then a division algebra D of degree pn over F is a cyclic algebra if and only if there is d ∈ D with dpn ∈ F – Fp. Let F(p) be the maximal p-extension of F. We show that F(p) has a noncyclic algebra of degree p if and only if a certain eigencomponent of the p-torsion of Br(F(p)(μp)) is nontrivial. To get a better understanding of F(p), we consider the valuations on F(p) with residue characteristic not p, and determine what residue fields and value groups can occur. Our results support the conjecture that the p torsion in Br(F(p)) is always trivial.

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

scholarly journals On the Automorphisms of the Four-dimensional Real Division Algebras

2017 ◽  
Vol 9 (2) ◽  
pp. 95
Author(s):  
Andre S. Diabang ◽  
Alassane Diouf ◽  
Mankagna A. Diompy ◽  
Alhousseynou Ba
Keyword(s):  
Division Algebra ◽  
Division Algebras ◽  
Automorphisms Groups

In this paper, we study partially the automorphisms groups of four-dimensional division algebra. We have proved that there is an equivalence between Der(A)=su(2) and Aut(A)=SO(3). For an unitary four-dimensional real division algebra, there is an equivalence between dim(Der(A))=1 and Aut(A)=SO(2).

scholarly journals Generalized Coinvariant Algebras for $G(r,1,n)$ in the Stanley-Reisner setting

10.37236/8109 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Daniël Kroes
Keyword(s):  
Polynomial Ring ◽  
Reflection Group ◽  
Roots Of Unity ◽  
Complex Reflection ◽  
Complex Reflection Group ◽  
Positive Integers ◽  
Coinvariant Algebra

Let $r$ and $n$ be positive integers, let $G_n$ be the complex reflection group of $n \times n$ monomial matrices whose entries are $r^{\textrm{th}}$ roots of unity and let $0 \leq k \leq n$ be an integer. Recently, Haglund, Rhoades and Shimozono ($r=1$) and Chan and Rhoades ($r>1$) introduced quotients $R_{n,k}$ (for $r>1$) and $S_{n,k}$ (for $r \geq 1$) of the polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ in $n$ variables, which for $k=n$ reduce to the classical coinvariant algebra attached to $G_n$. When $n=k$ and $r=1$, Garsia and Stanton exhibited a quotient of $\mathbb{C}[\mathbf{y}_S]$ isomorphic to the coinvariant algebra, where $\mathbb{C}[\mathbf{y}_S]$ is the polynomial ring in $2^n-1$ variables whose variables are indexed by nonempty subsets $S \subseteq [n]$. In this paper, we will define analogous quotients that are isomorphic to $R_{n,k}$ and $S_{n,k}$.

A Note on Subnormal Subgroups of Division Algebras

1978 ◽  
Vol 30 (01) ◽  
pp. 161-163 ◽  
Author(s):  
Gary R. Greenfield
Keyword(s):  
Division Algebra ◽  
Multiplicative Group ◽  
Subnormal Subgroup ◽  
Local Fields ◽  
Division Algebras ◽  
Subnormal Subgroups

Let D be a division algebra and let D* denote the multiplicative group of nonzero elements of D. In [3] Herstein and Scott asked whether any subnormal subgroup of D* must be normal in D*. Our purpose here is to show that division algebras over certain p-local fields do not satisfy such a “subnormal property”.

scholarly journals Universal and homogeneous embeddings of dual polar spaces of rank 3 defined over quadratic alternative division algebras

2016 ◽  
Vol 2016 (715) ◽  
Author(s):  
Bart De Bruyn ◽  
Hendrik Van Maldeghem
Keyword(s):  
Division Algebra ◽  
Division Algebras ◽  
Polar Spaces ◽  
Dual Polar Spaces

AbstractSuppose 𝕆 is an alternative division algebra that is quadratic over some subfield 𝕂 of its center

scholarly journals MAL'CEV–NEUMANN RINGS AND NONCROSSED PRODUCT DIVISION ALGEBRAS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250052 ◽  
Author(s):  
CÉCILE COYETTE
Keyword(s):  
Division Algebra ◽  
Laurent Series ◽  
Elementary Proof ◽  
Formal Series ◽  
Direct Approach ◽  
Division Algebras ◽  
Sufficient Condition ◽  
Series Ring

The first section of this paper yields a sufficient condition for a Mal'cev–Neumann ring of formal series to be a noncrossed product division algebra. This result is used in Sec. 2 to give an elementary proof of the existence of noncrossed product division algebras (of degree 8 or degree p2 for p any odd prime). The arguments are based on those of Hanke in [A direct approach to noncrossed product division algebras, thesis dissertation, Postdam (2001), An explicit example of a noncrossed product division algebra, Math. Nachr.251 (2004) 51–68, A twisted Laurent series ring that is a noncrossed product, Israel. J. Math.150 (2005) 199–2003].

Central simple algebras of prime exponent and divided power operations

2013 ◽  
Vol 11 (1) ◽  
pp. 113-123 ◽  
Author(s):  
A.S. Sivatski
Keyword(s):  
Division Algebra ◽  
Roots Of Unity ◽  
Central Simple Algebras ◽  
Divided Power ◽  
Open Questions ◽  
Primary Roots ◽  
Power Operations ◽  
Simple Algebras ◽  
Central Simple ◽  
Prime Exponent

AbstractLet p be a prime and F a field of characteristic different from p. Suppose all p-primary roots of unity are contained in F. Let α ∈ pBr(F) which has a cyclic splitting field. We prove that γi(α) = 0 for all i ≥ 2, where γi : pBr(F) → K2i(F)/pK2i(F) are the divided power operations of degree p. We also show that if char F ≠ 2, √−1 ∈ F*. D ∈2 Br(F), indD = 8 and a ∈ F* such that ind DF(√a) = 4, then γ3(D) = {a,s}γ2(D) for some s ∈ F*. Consequently, we prove that if D, considered as a division algebra, has a subfield of degree 4 of certain type, then γ3(D) = 0. At the end of the paper we pose a few open questions.

The derived length of the unit group of a group algebra — The case G′ = Sylp(G)

2016 ◽  
Vol 16 (08) ◽  
pp. 1750142 ◽  
Author(s):  
Tibor Juhász
Keyword(s):  
Nilpotent Group ◽  
Group Algebra ◽  
Upper Bound ◽  
Commutator Subgroup ◽  
Unit Group ◽  
Derived Length ◽  
Group Of Units

Let [Formula: see text] be an odd prime, and let [Formula: see text] be a nilpotent group, whose commutator subgroup is finite abelian satisfying [Formula: see text] and [Formula: see text]. In this contribution, an upper bound is given on the derived length of the group of units of the group algebra of [Formula: see text] over a field of characteristic [Formula: see text]. Furthermore, we show that this bound is achieved, whenever [Formula: see text] is cyclic.

scholarly journals Nicely semiramified division algebras over Henselian fields

2005 ◽  
Vol 2005 (4) ◽  
pp. 571-577 ◽  
Author(s):  
Karim Mounirh
Keyword(s):  
Division Algebra ◽  
Division Algebras ◽  
Central Division ◽  
Field Extensions ◽  
Finite Dimensional ◽  
Central Division Algebra ◽  
Radical Type ◽  
Maximal Subfield

This paper deals with the structure of nicely semiramified valued division algebras. We prove that any defectless finite-dimensional central division algebra over a Henselian fieldEwith an inertial maximal subfield and a totally ramified maximal subfield (not necessarily of radical type) (resp., split by inertial and totally ramified field extensions ofE) is nicely semiramified.

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