Editing Field extension (section)

== Generalizations ==
Field extensions can be generalized to [[ring extensions]] which consist of a [[ring (mathematics)|ring]] and one of its [[subring]]s. A closer non-commutative analog are [[central simple algebra]]s (CSAs) – ring extensions over a field, which are [[simple algebra]] (no non-trivial 2-sided ideals, just as for a field) and where the [[Center_(ring_theory)|center of the ring]] is exactly the field. For example, the only finite field extension of the real numbers is the complex numbers, while the quaternions are a central simple algebra over the reals, and all CSAs over the reals are [[Brauer equivalent]] to the reals or the quaternions. CSAs can be further generalized to [[Azumaya algebra]]s, where the base field is replaced by a commutative [[local ring]].