On the classification of central division algebras of linearly bounded degree over global fields and local fields (Q1320130)

An algebraic algebra \(A\) over a field \(K\) is called LBD (for linearly (or locally) bounded degree) if every finitely generated sub-vector space of \(A\) consists of elements of bounded degree. Algebraic algebras \(A\), \(B\) over \(K\) are called locally isomorphic if every locally finite-dimensional \(K\)-subalgebra of \(A\) of at most countable dimension is isomorphic to some \(K\)-subalgebra of \(B\) and vice versa. The purpose of the paper is to give the classification of all central division LBD-algebras over a global or local field \(K\) up to local \(K\)-isomorphism. In particular, locally finite-dimensional central division \(K\)-algebras of countable dimension are also classified up to \(K\)-isomorphism. This result extends the classical theorem of Brauer-Hasse-Noether and Albert which classifies the finite-dimensional central division algebras over a global field in terms of Hasse invariants. It turns out that every central division LBD-algebra \(A\) over a local or global field \(K\) is locally isomorphic to an infinite tensor product of central \(K\)- subalgebras of prime-power degree, whose Hasse invariants determine the algebra \(A\) up to local isomorphism. The main tool is an extension of the primary tensor product decomposition theorem to central division LBD- algebras over a field of arithmetic type.