On Köthe's normality question for locally finite-dimensional central division algebras

This paper considers Köthe’s question of whether every associative locally finite-dimensional (abbr., LFD) central division algebra $R$ over a field $K$ is a normally locally finite (abbr., NLF) algebra over $K$, that is, whether every nonempty finite subset $Y$ of $R$ is contained in a finite-dimensional central $K$-subalgebra $\mathcal{R} _{Y}$ of $R$. It shows that the answer to the posed question is negative if $K$ is a purely transcendental extension of infinite transcendence degree over an algebraically closed field $k$. On the other hand, central division LFD-algebras over $K$ turn out to be NLF in the following special cases: (i) $K$ is a finitely-generated extension of a finite or a pseudo-algebraically closed perfect field $K _{0}$; (ii) $K$ is a higher-dimensional local field with last residue field equal to $K _{0}$.

💡 Research Summary

The paper addresses a long‑standing problem posed by Köthe: whether every associative locally finite‑dimensional (LFD) central division algebra R over a field K must be normally locally finite (NLF), i.e., whether each finite subset Y⊂R is contained in a finite‑dimensional central K‑subalgebra. The authors give a definitive answer that depends on the nature of the base field.

First, they construct a counterexample showing that the answer is negative in general. Let K₀ be an algebraically closed field and consider a purely transcendental extension K = K₀(xₙ, yₙ : n∈ℕ) of infinite transcendence degree. Fix a prime p≠char(K₀). For each n they build a finite‑dimensional central division algebra Rₙ over the intermediate field Kₙ = K₀(x₁,…,xₙ, y₁,…,yₙ) such that the centre Zₙ is a Galois extension of Kₙ with abelian Galois group of order pⁿ and period p. The algebras are arranged so that Zₙ∩Z_{2n}=Kₙ. The union R = ⋃ₙRₙ is then a central division LFD‑algebra over K whose only finite‑dimensional central K‑subalgebra is K itself. Consequently R is not NLF, establishing a negative answer to Köthe’s question for fields that are purely transcendental extensions of infinite transcendence degree over an algebraically closed base (Theorem 2.1, Proposition 2.4).

The second major contribution is a positive result for a large class of fields, called ΦBr‑fields (Brauer‑finite‑dimensional fields). A field F belongs to ΦBr if for each prime p there exists an integer m(p) such that ind(A) divides exp(A)^{m(p)} for every finite‑dimensional central simple algebra A over F. This class includes global fields, ℓ‑adic fields, and, crucially for the paper, fields that are finitely generated over a finite or pseudo‑algebraically closed (PAC) perfect field, as well as higher‑dimensional local fields with a finite last residue field.

Theorem 3.1 shows that if F∈ΦBr and R is a central division LFD‑algebra over F, then R is NLF. Moreover R admits a canonical decomposition \