Division Algebras Of Prime Degree Research Articles

Connective $K$-theory and Adams operations

We investigate the relations between the Grothendieck group of coherent modules of an algebraic variety and its Chow group of algebraic cycles modulo rational equivalence. Those are in essence torsion phenomena, which we attempt to control by considering the action of the Adams operations on the Brown-Gersten-Quillen spectral sequence and related objects, such as connective K_0-theory. We provide elementary arguments whenever possible. As applications, we compute the connective K_0-theory of the following objects: (1) the variety of reduced norm one elements in a central division algebra of prime degree; (2) the classifying space of the split special orthogonal group of odd degree.

A note on the existence of cyclic algebras in division algebras

ABSTRACTLet F be a Henselian field such that is a global field. It is shown that if D is an F-central division algebra such that then D contains a cyclic division algebra of prime degree even though D can be a noncrossed product. Here, our approach is based on the Albert’s cyclicity criterion and centralizer theorem.

Characters of the norm-one units of local division algebras of prime degree

We give an explicit construction of all complex continuous irreducible characters of the group SL1(D), where D is a division algebra of prime degree ℓ over a local field of odd residual characteristic different from ℓ. For ℓ odd, we show that all such characters of SL1(D) are induced from linear characters of compact-open subgroups of SL1(D). We also compute an explicit formula for the representation zeta function of SL1(D).

Division algebras of prime degree with infinite genus

The genus gen(D) of a finite-dimensional central division algebra D over a field F is defined as the collection of classes [D′] ∈ Br(F), where D′ is a central division F-algebra having the same maximal subfields as D. For any prime p, we construct a division algebra of degree p with infinite genus. Moreover, we show that there exists a field K such that there are infinitely many nonisomorphic central division K-algebras of degree p and any two such algebras have the same genus.

On the motive of the group of units of a division algebra

AbstractWe 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.

Division Algebras of Prime Degree and Maximal Galois p-Extensions

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.

Galois cohomology of fields without roots of unity

There is a standard correspondence between elements of the cohomology group H 1( F, μ n ) (with the trivial action of Γ F =Gal( F s/ F) on μ n ) and cyclic extensions of dimension n over F. We extend this to a correspondence between the cohomology groups H 1( F, μ n ) where the action of Γ F on μ n varies, and the extensions of dimension n of K which are Galois over F, where K= F[ μ n ] and [ K: F] is prime to n. The cohomology groups are also related to eigenspaces of H 1(K, Z/n) with respect to the natural action of Gal( K/ F). As a result, we extend Albert's cyclicity criterion, stated in the 1930s for division algebras of prime degree, to algebras of prime-power degree over F, under the assumption stated above. We also extend the Rosset–Tate result on the corestriction of cyclic algebras in the presence of roots of unity, to extensions in which roots of unity live in an extension of dimension ⩽3 over the base field. In particular, if roots of unity live in a quadratic extension of the base field, then corestriction of a cyclic algebra along a quadratic extension is similar to a product of two cyclic algebras. Another application is that F-central algebras which are split by a certain semidirect product extension of F, are cyclic. In particular, if [ K: F]=2 then algebras over F which are split by an odd order dihedral extension, are cyclic. We also construct generic examples of algebras which become cyclic after extending scalars by roots of unity, and show the existence of elements for which most powers have reduced trace zero.

Cyclicity conditions for division algebras of prime degree

Let D D be a division algebra of prime degree p p . A set of criteria is given for cyclicity of D D in terms of subgroups of the multiplicative group D ∗ D^* of D D . It is essentially shown that D D is cyclic if and only if D ∗ D^* contains a nonabelian metabelian subgroup.

Cyclicity of Division Algebras of Prime Degree

Cyclicity of Division Algebras of Prime Degree

Cyclicity of division algebras of prime degree

It is not known whether a central division algebra D of prime degree p over a field F can be noncyclic. Every such algebra which has been constructed is cyclic and, when any of various mild conditions is imposed on D, it can be proved that D is cyclic. We consider only the case in which the characteristic of D is equal to its degree p. For this case, it is known' that cyclicity of D is guaranteed by the following conditions: there is an extension field K of F such that (i) K/F has degree m with (m, p) = 1; (ii) DK is cyclic with maximal subfield W which is galois over F; (iii) K/F is cyclic. We shall prove:

On Associative Division Algebras of Prime Degree

On Associative Division Algebras of Prime Degree

On associative division algebras of prime degree

In 1938 Richard Brauer showed2 that, if Z is an associative division algebra of degree five over its center j, there is a field 9 of degree at most twelve over j such that the scalar extension Z X 9 is a cyclic algebra over R. Indeed 9 = Q(aj, a2, (X3), where a, is a root of a quadratic equation over j, a2 is a root of a quadratic equation over (aj), and a3 is a root of a cubic equation over ?(aj, ax2). Since that time there has been no progress in the study of the structure of associative central division algebras of prime degree. In view of Brauer's result it seems reasonable, as a first step in the study of central division algebras Z of prime degree p over j, to consider the case where there is a quadratic field 9 over j such that Za is cyclic. Then Za = Z X has a subfield <3 which is cyclic of degree p over R. The simplest subcase is that where ?3 is actually normal, but, of course, not cyclic over j. We shall treat this case under the assumption that j has characteristic p, and shall prove that then Z is a cyclic algebra over . Our proof proceeds as follows. We are assuming that Z is a central division algebra of prime degree p over j and that 9 = j(t) is a quadratic extension of j. Let J be the automorphism of 9 such that ti = t. Then we may extend J to an automorphism J of Z X 9 such that dJ=d for every d in Z. We are also assuming that there is an element z in Z X 9 such that the field 3 = R (z) is not only cyclic over

A note on normal division algebras of prime degree

A note on normal division algebras of prime degree