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'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple a ...
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'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple a ...
over a field ''F''. Then ''A'' is said to be cyclic if it contains a
strictly maximal subfield In algebra, a subfield of an algebra ''A'' over a field ''F'' is an ''F''- subalgebra that is also a field. A maximal subfield is a subfield that is not contained in a strictly larger subfield of ''A''.
If ''A'' is a finite-dimensional central sim ...
''E'' such that ''E''/''F'' is a
cyclic field extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable ...
(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 group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
).
See also
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Factor system#Cyclic algebras - cyclic algebras described by factor systems.
*
Brauer group#Cyclic algebras - cyclic algebras are representative of Brauer classes.
References
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Algebra