We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings D. We define their fixed points to be the points $\lambda \in D$ for which $f^{\circ n}(\lambda )=\lambda $ for any $n \in \mathbb {N}$ , where $f^{\circ n}(x)$ is defined recursively by $f^{\circ n}(x)=f(f^{\circ (n-1)}(x))$ and $f^{\circ 1}(x)=f(x)$ . Periodic points are similarly defined. We prove that $\lambda $ is a fixed point of $f(x)$ if and only if $f(\lambda )=\lambda $ , which enables the use of known results from the theory of polynomial equations, to conclude that any polynomial of degree $m \geq 2$ has at most m conjugacy classes of fixed points. We also show that in general, periodic points do not behave as in the commutative case. We provide a sufficient condition for periodic points to behave as expected.

Let $F$ be a non-Archimedean local field, and let $G^{\sharp }$ be the group of $F$-rational points of an inner form of $\text{SL}_{n}$. We study Hecke algebras for all Bernstein components of $G^{\sharp }$, via restriction from an inner form $G$ of $\text{GL}_{n}(F)$.

For any packet of L-indistinguishable Bernstein components, we exhibit an explicit algebra whose module category is equivalent to the associated category of complex smooth $G^{\sharp }$-representations. This algebra comes from an idempotent in the full Hecke algebra of $G^{\sharp }$, and the idempotent is derived from a type for $G$. We show that the Hecke algebras for Bernstein components of $G^{\sharp }$ are similar to affine Hecke algebras of type $A$, yet in many cases are not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group.

We consider the algebraic group GL1 (A), where A is a division algebra of prime degree over a field F, and the associated motive in the Voevodsky category of motivic complexes (F). We relate the motive of GL1 (A) to the motive of the Čech simplicial scheme χ, associated to the Severi-Brauer variety of A, and compute the second differential in the resulting spectral sequence for motivic cohomology.

Bushnell and Kutzko gave a complete and effective classification of the smooth dual of $\mbox{GL}(N,F)$, where $F$ is a non-archimedean local field. Similarly, Zink gave a classification of the smooth dual of $D^{\times}$, where $D$ is a division algebra with centre $F$, in terms of non-canonical objects and under the restrictive hypothesis that $F$ has characteristic $0$. In this paper, we extend part of Bushnell and Kutzko's formalism to $D^{\times}$ and obtain a complete classification of the smooth dual working for any characteristic. The crucial point of this work is to define a good way of splitting the algebra $D$ so that the important notion of {\it simple stratum}, and its properties, can be translated to $D^{\times}$ by some descent arguments.

1991 Mathematics Subject Classification: 12E15, 20G05, 20G25, 22E50.

We study a ramification theory for a division algebra D of the following type: The center of D is a complete discrete valuation field K with the imperfect residue field F of certain type, and the residue algebra of D is commutative and purely inseparable over F. Using Swan conductors of the corresponding element of Br(K), we define ψ-function of D/K, and it describe the action of the reduced norm map on the filtration of D$^*$. We also gives a relation among the Swan conductors and the ’ramification number‘ of D, which is defined by the commutator group of D$^*$.

We study a ramification theory for a division algebra D of the following type: The center of D is a complete discrete valuation field K with the imperfect residue field F of certain type, and the residue algebra of D is commutative and purely inseparable over F. Using Swan conductors of the corresponding element of Br(K), we define Herbrand‘s ψ-function of D/K, and it describes the action of the reduced norm map on the filtration of D$^*$. We also gives a relation among the Swan conductors and the ’ramification number‘ of D, which is defined by the commutator group of D$^*$.

We show that every element in the Brauer-Wall group of a field with characteristic different from 2 is represented uniquely by a division graded algebra, (i.e. homogeneous elements are invertible) but, of course, not necessarily by a graded (division algebra). This is a fairly direct consequence of Wall's structure theory for central simple Z/2-graded algebras.

In this note it is proved that a (real or complex) semiprime Banach algebra A satisfying xAx = x2Ax2 for every x ∈ A is a direct sum of a finite number of division Banach algebras.

In this note we show that taking a scalar extension of two elementarily equivalent finite-dimensional algebras over the same field preserves elementary equivalence. The general question of whether or not tensor product preserves elementary equivalence was originally raised in [4]. In [3] Feferman relates an example of Ersov which answers the question negatively. Eklof and Olin [7] also provide a counterexample to the general question in the context of two-sorted structures. Thus the result proved below is a partial positive answer to a general question whose status has been resolved negatively. From the viewpoint of applied model theory it seems desirable to find contexts in which positive statements of preservation can be obtained. Our result does have an application; a corollary to it increases our understanding of what it means for two division algebras to be elementarily equivalent.

All algebras are finite-dimensional algebras over fields. All algebras contain an identity element, but are not necessarily associative.

Recall that the center of a not necessarily associative algebra A is the set of elements which commute and “associate” with all elements of A. The notion of a scalar extension is an important one in algebra. If A is an algebra over F and G is an extension field of F, then the scalar extension of A by G is the algebra A ⊗F G.

Let D be a division algebra whose class [D] is in B(K), the Brauer group of an algebraic number field K. If [D⊗KL] is the trivial class in B(L), then we say that L is a splitting field for D or L splits D. The splitting fields in D of smallest dimension are the maximal subfields of D. Although there are infinitely many maximal subfields of D which are cyclic extensions of K; from the perspective of the Schur Subgroup S(K) of B(K) the natural splitting fields are the cyclotomic ones. In (Cyclotomic Splitting Fields, Proc. Amer. Math. Soc. 25 (1970), 630-633) there are errors which have led to the main result of this paper, namely to provide necessary and sufficient conditions for (D) in S(K) to have a maximal subfield which is a cyclic cyclotomic extension of K, a finite abelian extension of Q. A similar result is provided for quaternion division algebras in B(K).