In algebra, a subfield of an

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

''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 operat ...

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'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple a ...

, 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, {{algebra-stub Algebra