In algebra, the Jacobson–Bourbaki theorem is a theorem used to extend
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
to
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
s that need not be separable. It was introduced by for commutative
fields and extended to non-commutative fields by , and who credited the result to unpublished work by
Nicolas Bourbaki
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook i ...
. The extension of Galois theory to
normal extension
In abstract algebra, a normal extension is an algebraic field extension ''L''/''K'' for which every irreducible polynomial over ''K'' which has a root in ''L'', splits into linear factors in ''L''. These are one of the conditions for algebraic e ...
s is called the Jacobson–Bourbaki correspondence, which replaces the correspondence between some
subfields of a field and some subgroups of a
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 po ...
by a correspondence between some sub division rings of a
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
and some
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 ...
s of an associative algebra.
The Jacobson–Bourbaki theorem implies both the usual Galois correspondence for subfields of a Galois extension, and Jacobson's Galois correspondence for subfields of a
purely inseparable extension In algebra, a purely inseparable extension of fields is an extension ''k'' ⊆ ''K'' of fields of characteristic ''p'' > 0 such that every element of ''K'' is a root of an equation of the form ''x'q'' = ''a'', wit ...
of exponent at most 1.
Statement
Suppose that ''L'' is a
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
.
The Jacobson–Bourbaki theorem states that there is a natural 1:1 correspondence between:
*Division rings ''K'' in ''L'' of finite index ''n'' (in other words ''L'' is a finite-dimensional left vector space over ''K'').
*Unital ''K''-algebras of finite dimension ''n'' (as ''K''-vector spaces) contained in the ring of endomorphisms of the additive group of ''K''.
The sub division ring and the corresponding subalgebra are each other's commutants.
gave an extension to sub division rings that might have infinite index, which correspond to closed subalgebras in the finite topology.
References
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{{DEFAULTSORT:Jacobson-Bourbaki theorem
Field (mathematics)
Theorems in algebra