Generic abelian crossed products and p-algebras

Open problems on central simple algebras

“…It is natural to ask whether such elements exist in algebras of degree greater than p. When p 2, the degree is 4, and the characteristic is not 2, Albert's crossed product result shows that they do exist. On the other hand, it is shown in [AS78] that there exist algebras of degree p 2 and characteristic p with no p-central elements. Using this, it is shown in [Sal80] that for n a multiple of p 2 , the universal division algebra of degree n over the rational field Q has no p-central elements.…”

“…9.31]. Louis Isaac Gordon proved the existence of noncyclic 2-algebras (of period and index 4) in [Gor40] and Amitsur and Saltman proved the existence of (generic) non-cyclic division p-algebras of every degree p n (n 2) in [AS78]. These results raised questions about bounds of cyclic splitting fields.…”

“…Moreover, in case F has prime characteristic p, Albert proved that every p-algebra (i.e., central simple algebra of period a power of p) is Brauer-equivalent to a cyclic algebra. But [AS78] gives an example of a noncyclic p-algebra. One can lift this example to a complete discrete valuation ring of characteristic 0 and so find a central simple algebra A over a field of characteristic 0 such that M n ÔAÕ is cyclic for some n but A is not.…”

Division algebras of degree 4 and 8 with involution

“…The main idea is to use the generic abelian crossed products of [2], modified slightly to account for the presence of an involution. Suppose R is an abelian crossed product, i.e.…”

Division algebras over 𝐶₂- and 𝐶₃-fields

“…Remark 1.2. One can refine this result by means of the Amitsur-Saltman construction [3] of an arbitrary central simple algebra of degree 4 in terms of a maximal subfield K Galois over the center F . We can write K = F (a 1 , a 2 ) where a 2 i ∈ F, and one has z i ∈ K such that z i a i z…”