Cyclic algebra

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

• Factor system § Cyclic algebras – cyclic algebras described by factor systems.
• Brauer group § Cyclic algebras – cyclic algebras are representative of Brauer classes.

References

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