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In ring theory and related areas of mathematics a central simple algebra (CSA)
over a field K is a finite-dimensional associative K-algebra A,
which is simple, and for which the center is exactly K.
As an example, note that any simple algebra is a central simple algebra over its center.

For example, the complex numbers C form a CSA over themselves,
but not over the real numbers R (the center of C is all of C, not just R).
The quaternions H form a 4-dimensional CSA over R,
and in fact represent the only non-trivial element of
the Brauer group of the reals.

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