Cyclic algebra explained

In algebra, a cyclic division algebra is one of the basic examples of a division algebra over a field and plays a key role in the theory of central simple algebras.

Definition

Let A be a finite-dimensional central simple algebra over a field F. Then A is said to be cyclic if it contains a strictly maximal subfield E such that E/F is a cyclic field extension (i.e., the Galois group is a cyclic group).

See also

• cyclic algebras described by factor systems.
• cyclic algebras are representative of Brauer classes.

References

• Book: Pierce, Richard S.. Associative Algebras. Springer-Verlag. 1982. 978-0-387-90693-5. Graduate Texts in Mathematics, volume 88. 249353240. registration.
• Book: Weil, André. André Weil. Basic Number Theory. third. Springer. 1995. 978-3-540-58655-5. 32381827.