Noncountable normally locally finite division algebras

Abstract

A (commutative) field F is regular (see [1 ] the bibliography) if it is not finite, and if in addition it is true that the direct (= Kronecker =tensor) product two normal (=central) division algebras, finite orders, over F is not a division algebra unless their orders are relatively prime; algebraic number fields and p-adic fields are examples regular fields. A division algebra 2 over a field F is normally locallyfinite if any finite subset 2 is contained in a normal (over F) division sub-algebra 2 finite order; in [1], such algebras were called of type 1. A subisomorphism an algebra 2 over F is an algebra-isomorphism 2 into f, and it is proper if it is not onto. If 2( is a normally locally finite division algebra over the regular field F, without a finite basis over F, a characteristic sub-algebra 2 is any normally locally finite division sub-algebra Z XI, with countably infinite basis over F, having the property that any normally locally finite division sub-algebra 2, with finite or countable basis over F, is isomorphic to a sub-algebra Z. It was proved in [1] that any 2 the previous type has a characteristic sub-algebra, unique but for isomorphisms; it was also proved that there exists a normally locally finite division algebra over the regular field F, with infinite noncountable basis, and with a given characteristic sub-algebra Z, if and only if Z admits proper subisomorphisms; [1 ] contains a rather involved proof the fact that any Z admits proper subisomorphisms if F is not countable, and thus establishes the existence normally locally finite division algebras, with infinite noncountable basis, over any noncountable regular field; this seems to be the only known example such algebras. We shall present here a very simple proof the same result, and will, at the same time, dispense with the condition noncountability F.