Division algebras with infinite genus

Abstract

We give an explicit construction of division algebras with infinite genus. We go on to show that there exists a field K for which (1) there are infinitely many nonisomorphic quaternion division algebras with center K, and (2) any two quaternion division algebras with center K are pairwise weakly isomorphic. In fact, we show that there are infinitely many nonisomorphic fields satisfying these two conditions.

The finiteness of the genus of a finite-dimensional division algebra, and some generalizations

“…While the latter notion may be useful, some caution needs to be exercised in interpreting its consequences: for example, we don't know whether the fact that the division D and D ′ have the same maximal subfields implies that the matrix algebras M ℓ (D) and M ℓ (D ′ ) have the same maximal subfield /étale subalgebras for any (or even some) ℓ > that the assumption that K is finitely generated is essential as the genus of a division algebra over an infinitely generated field can be infinite, cf. [19], [35].) The proof of Theorem 1.1 also uses the analysis of ramification, but bypasses some technical arguments developed in [7], [9].…”

On the size of the genus of a division algebra

“…It should be noted that the assumption of finite generation of K is essential for the finiteness of the genus -see [30] and [48] for a construction of division algebras with infinite genus in the general situation.…”

Division algebras of prime degree with infinite genus

“…A variation of this problem is concerned only with the consideration of the collection of maximal subfields of algebras. Recently this has been studied by several authors (see [2], [3], [4], [6], [7]). 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. This means that D and D ′ have the same degree n, and a field extension K/F of degree n admits an F -embedding K ֒→ D if and only if it admits an F -embedding K ֒→ D ′ .…”

“…In [6], it is shown that there are quaternion algebras with infinite genus. Besides, it is proved that there exists a field F over which there are infinitely many nonisomorphic quaternion algebras with center F , and any two quaternion division algebras with center F have the same genus.…”

“…In this paper, borrowing some ideas from [6] and [8], we generalize the results from [6] to the case of division algebras of any prime degree. More precisely, for any prime p, we construct a division algebra of degree p with infinite genus (see Theorem 4 below).…”