Symbol length in the Brauer group of a field

Abstract

Abstract. We bound the symbol length of elements in the Brauer group of a field K containing a C m field (for example any field containing an algebraically closed field or a finite field), and solve the local exponent-index problem for a C m field F . In particular, for a C m field F , we show that every F central simple algebra of exponent p t is similar to the tensor product of at most len(p t , F ) ≤ t(p m−1 − 1) symbol algebras of degree p t . We then use this bound on the symbol length to show that the index of such algebras is bounded by (p t ) (p m−1 −1) , which in turn gives a bound for any algebra of exponent n via the primary decomposition. Finally for a field K containing a C m field F , we show that every F central simple algebra of exponent p t and degree p s is similar to the tensor product of at most len(p t , p s , K) ≤ len(p t , L) symbol algebras of degree p t , where L is a C m+edL(A)+p s−t −1 field.

Kato–Milne cohomology and polynomial forms

“…Recall that when char(F) = p, p Br(F) H 2 p (F). It was shown in [Cha17, Corollary 3.3] that if the maximal dimension of an anisotropic form of degree p over F is d then the symbol length of p Br(F) is bounded from above by d−1 p − 1, providing a characteristic p analogue to a similar result obtained in [Mat16] in the case of char(F) p. As a result, if F is C m then d p m and so this upper bound boils down to p m−1 − 1. However, the symbol length of p Br(F) when F is a C m field with char(F) = p is bounded from above by the p-rank which is at most m (see [Alb68,Theorem 28]).…”

Essential Dimension, Symbol Length and -rank

“…We conclude with some comments on the case of an algebraically closed field k of characteristic prime to p. Given a field E of transcendence degree r over k, E is a C r field, and so the symbol length of a central simple algebra A in Br p m (E) of degree p ℓm is at most m(p r−1 − 1) by [15,Theorem 8.2]. Therefore, if we start with a central simple algebra A of degree p ℓm and exponent p m over a field F containing k whose essential dimension is r, then sl p m ([A]) m(p r−1 − 1) (a formula which was already implicitly obtained in [15,Section 5]). By solving for r, we obtain the formula Excluding the case of p = m = ℓ = 2, the existence of indecomposable algebras A of degree p ℓm and exponent p m provides algebras A of symbol length at least ℓ + 1, which gives the bound ed(Alg p ℓm ,p m ) 1 + log p ℓ + 1 m + 1 .…”

Noncyclic division algebras over fields of Brauer dimension one

“…The Brauer dimension of the function field of a p-adic curve was proved to be two in [22], and shown to have cyclic length two in [8]. See [13], [18], and [14] for known results on the relationship between Brauer dimension and symbol length.…”