In algebra, a cyclic division algebra is one of the basic examples of a
division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
over a field and plays a key role in the theory of
central simple algebra
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'' that is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple ...
s.
Definition
Let ''A'' be a finite-dimensional
central simple algebra
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'' that is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple ...
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
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
is a
cyclic group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
).
See also
* cyclic algebras described by factor systems.
* cyclic algebras are representative of Brauer classes.
References
*
*
Algebras
Ring theory
{{algebra-stub