Difference between revisions of "Jacobson-Bourbaki theorem"
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Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001024.png" /> be a subfield of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001025.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001026.png" /> is a finite-dimensional vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001027.png" /> and to each subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001028.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001029.png" /> let correspond the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001030.png" /> of linear operators on the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001031.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001032.png" />. The Jacobson–Bourbaki theorem asserts that this correspondence is a [[Bijection|bijection]] between the subfields of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001033.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001034.png" /> and the set of subrings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001035.png" /> that are left vector spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001036.png" /> of finite dimension. Moreover, the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001037.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001038.png" /> equals the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001039.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001040.png" />. | Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001024.png" /> be a subfield of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001025.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001026.png" /> is a finite-dimensional vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001027.png" /> and to each subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001028.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001029.png" /> let correspond the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001030.png" /> of linear operators on the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001031.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001032.png" />. The Jacobson–Bourbaki theorem asserts that this correspondence is a [[Bijection|bijection]] between the subfields of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001033.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001034.png" /> and the set of subrings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001035.png" /> that are left vector spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001036.png" /> of finite dimension. Moreover, the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001037.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001038.png" /> equals the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001039.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001040.png" />. | ||
− | This theorem has been used by N. Jacobson to develop a Galois theory of finite, purely inseparable field extensions of exponent one, in which groups of automorphisms are replaced by Lie algebras of derivations, [[#References|[a1]]]. | + | This theorem has been used by N. Jacobson to develop a Galois theory of finite, [[Purely inseparable extension|purely inseparable]] field extensions of exponent one, in which groups of automorphisms are replaced by Lie algebras of derivations, [[#References|[a1]]]. |
The Jacobson–Bourbaki theorem can be regarded as an instance of Morita duality. From the theory of [[Morita equivalence|Morita equivalence]] the following very general result is obtained. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001041.png" /> be a [[Ring|ring]], let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001042.png" /> be the ring of endomorphisms of the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001043.png" /> an note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001044.png" /> is a left [[Module|module]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001045.png" />. There is a one-to-one correspondence between those subrings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001046.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001047.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001048.png" /> is a finitely generated projective generator in the [[Category|category]] of right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001049.png" />-modules and the subrings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001050.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001051.png" /> is a submodule of the left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001052.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001054.png" /> is a finitely generated projective generator in the category of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001055.png" />-modules. | The Jacobson–Bourbaki theorem can be regarded as an instance of Morita duality. From the theory of [[Morita equivalence|Morita equivalence]] the following very general result is obtained. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001041.png" /> be a [[Ring|ring]], let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001042.png" /> be the ring of endomorphisms of the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001043.png" /> an note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001044.png" /> is a left [[Module|module]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001045.png" />. There is a one-to-one correspondence between those subrings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001046.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001047.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001048.png" /> is a finitely generated projective generator in the [[Category|category]] of right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001049.png" />-modules and the subrings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001050.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001051.png" /> is a submodule of the left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001052.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001054.png" /> is a finitely generated projective generator in the category of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001055.png" />-modules. |
Revision as of 17:15, 7 November 2016
The Jacobson–Bourbaki theorem is especially useful for generalizations of the Galois theory of finite, normal and separable field extensions.
Let be a finite group of automorphisms of a field
, and let
be the subfield of elements of
that are invariant under the action of
. Then
is a normal and separable extension of
(cf. Extension of a field), and there is a one-to-one correspondence between the subfields of
containing
and the subgroups of
(cf. also Galois theory). The elements of
are linear operators on the vector space
over
; by the operation of multiplication, the elements of
can be represented as linear operators on
over
(the regular representation), and the ring
of all linear operators on
over
is generated by
and
; indeed, it is the cross product of
and
.
Now, let be a subfield of
such that
is a finite-dimensional vector space over
and to each subfield
containing
let correspond the ring
of linear operators on the vector space
over
. The Jacobson–Bourbaki theorem asserts that this correspondence is a bijection between the subfields of
containing
and the set of subrings of
that are left vector spaces over
of finite dimension. Moreover, the dimension of
over
equals the dimension of
over
.
This theorem has been used by N. Jacobson to develop a Galois theory of finite, purely inseparable field extensions of exponent one, in which groups of automorphisms are replaced by Lie algebras of derivations, [a1].
The Jacobson–Bourbaki theorem can be regarded as an instance of Morita duality. From the theory of Morita equivalence the following very general result is obtained. Let be a ring, let
be the ring of endomorphisms of the additive group of
an note that
is a left module over
. There is a one-to-one correspondence between those subrings
of
such that
is a finitely generated projective generator in the category of right
-modules and the subrings
such that
is a submodule of the left
-module
and
is a finitely generated projective generator in the category of left
-modules.
References
[a1] | N. Jacobson, "Galois theory of purely inseparable fields of exponent one" Amer. J. Math. , 66 (1944) pp. 645–648 |
Jacobson-Bourbaki theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobson-Bourbaki_theorem&oldid=22607