In algebra, a subfield of an

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

''A'' over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''F'' is an ''F''-

subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear opera ...

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 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 ...

, then a subfield ''E'' of ''A'' is called a strictly maximal subfield if

= (\dim_F A)^

References

* Richard S. Pierce. ''Associative algebras''. Graduate texts in mathematics, Vol. 88, Springer-Verlag, 1982, Abstract algebra {{algebra-stub