In algebra, a subfield of an
algebra
Algebra () is one of the areas of mathematics, 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 mathem ...
''A'' over a
field ''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 oper ...
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 simp ...
, then a subfield ''E'' of ''A'' is called a strictly maximal subfield if
.
References
* Richard S. Pierce. ''Associative algebras''. Graduate texts in mathematics, Vol. 88, Springer-Verlag, 1982,
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Algebra