  We give a necessary and sufficient condition for an atomless Boolean algebra
to be countably generated, and use it to give new proofs of some some know
facts due to Gaifman-Hales and Solovay and also due to Jech, Kunen and Magidor.
We also show that Jensen's coding theorem can be used to provide cardinal
preserving countably generated complete Boolean algebras of arbitrary large
cardinality. This answers a question of Jech [5] from 1976.
