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The Jacobson–Bourbaki theorem is especially useful for generalizations of the Galois theory of finite, normal and separable field extensions.
 
The Jacobson–Bourbaki theorem is especially useful for generalizations of the Galois theory of finite, normal and separable field extensions.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j1100101.png" /> be a [[Finite group|finite group]] of automorphisms of a [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j1100102.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j1100103.png" /> be the subfield of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j1100104.png" /> that are invariant under the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j1100105.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j1100106.png" /> is a normal and separable extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j1100107.png" /> (cf. [[Extension of a field|Extension of a field]]), and there is a one-to-one correspondence between the subfields of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j1100108.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j1100109.png" /> and the subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001010.png" /> (cf. also [[Galois theory|Galois theory]]). The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001011.png" /> are linear operators on the [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001012.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001013.png" />; by the operation of multiplication, the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001014.png" /> can be represented as linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001015.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001016.png" /> (the [[Regular representation|regular representation]]), and the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001017.png" /> of all linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001018.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001019.png" /> is generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001021.png" />; indeed, it is the [[Cross product|cross product]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j110/j110010/j11001023.png" />.
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Let $G$ be a [[finite group]] of automorphisms of a [[field]] $P$, and let $F$ be the subfield of elements of $P$ that are invariant under the action of $G$. Then $P$ is a normal and separable extension of $F$ (cf. [[Extension of a field]]), and there is a one-to-one correspondence between the subfields of $P$ containing $F$ and the subgroups of $G$ (cf. also [[Galois theory]]). The elements of $G$ are linear operators on the [[vector space]] $P$ over $F$; by the operation of multiplication, the elements of $P$ can be represented as linear operators on $P$ over $F$ (the [[regular representation]]), and the ring $\text{Hom}_F(P,P)$ of all linear operators on $P$ over $F$ is generated by $P$ and $G$; indeed, it is the [[cross product]] of $P$ and $G$.
  
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" />.
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Now, let $F$ be a subfield of $P$ such that $P$ is a finite-dimensional vector space over $F$ and to each subfield $E \subseteq P$ containing $F$ let correspond the ring $\text{Hom}_E(P,P)$ of linear operators on the vector space $P$ over $E$. The Jacobson–Bourbaki theorem asserts that this correspondence is a [[bijection]] between the subfields of $P$ containing $F$ and the set of subrings of $\text{Hom}_F(P,P)$ that are left vector spaces over $P$ of finite dimension. Moreover, the dimension of $P$ over $E$ equals the dimension of $\text{Hom}_E(P,P)$ over $P$.
  
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]]].
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This theorem has been used by N. Jacobson to develop a Galois theory of finite, [[Purely inseparable extension|purely inseparable]] field extensions of [[exponent of a purely inseparable extension|exponent]] one, in which groups of automorphisms are replaced by [[Lie algebra]]s of [[Derivation in a ring|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.
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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 $P$ be a [[ring]], let $\text{End}(P,{+})$ be the ring of endomorphisms of the additive group of $P$ and note that $\text{End}(P,{+})$ is a left [[module]] over $P$. There is a one-to-one correspondence between those subrings $E$ of $P$ such that $P$ is a finitely generated [[Generator of a category|projective generator]] in the [[category]] of right $E$-modules and the subrings $L \subseteq \text{End}(P,{+})$ such that $L$ is a submodule of the left $P$-module $\text{End}(P,{+})$ and $P$ is a finitely generated projective generator in the category of left $L$-modules.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Jacobson,  "Galois theory of purely inseparable fields of exponent one"  ''Amer. J. Math.'' , '''66'''  (1944)  pp. 645–648</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Jacobson,  "Galois theory of purely inseparable fields of exponent one"  ''Amer. J. Math.'' , '''66'''  (1944)  pp. 645–648 {{DOI|10.2307/2371772}}  {{ZBL|0063.03019}}</TD></TR>
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</table>
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{{TEX|done}}

Latest revision as of 19:10, 9 November 2016

The Jacobson–Bourbaki theorem is especially useful for generalizations of the Galois theory of finite, normal and separable field extensions.

Let $G$ be a finite group of automorphisms of a field $P$, and let $F$ be the subfield of elements of $P$ that are invariant under the action of $G$. Then $P$ is a normal and separable extension of $F$ (cf. Extension of a field), and there is a one-to-one correspondence between the subfields of $P$ containing $F$ and the subgroups of $G$ (cf. also Galois theory). The elements of $G$ are linear operators on the vector space $P$ over $F$; by the operation of multiplication, the elements of $P$ can be represented as linear operators on $P$ over $F$ (the regular representation), and the ring $\text{Hom}_F(P,P)$ of all linear operators on $P$ over $F$ is generated by $P$ and $G$; indeed, it is the cross product of $P$ and $G$.

Now, let $F$ be a subfield of $P$ such that $P$ is a finite-dimensional vector space over $F$ and to each subfield $E \subseteq P$ containing $F$ let correspond the ring $\text{Hom}_E(P,P)$ of linear operators on the vector space $P$ over $E$. The Jacobson–Bourbaki theorem asserts that this correspondence is a bijection between the subfields of $P$ containing $F$ and the set of subrings of $\text{Hom}_F(P,P)$ that are left vector spaces over $P$ of finite dimension. Moreover, the dimension of $P$ over $E$ equals the dimension of $\text{Hom}_E(P,P)$ over $P$.

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 $P$ be a ring, let $\text{End}(P,{+})$ be the ring of endomorphisms of the additive group of $P$ and note that $\text{End}(P,{+})$ is a left module over $P$. There is a one-to-one correspondence between those subrings $E$ of $P$ such that $P$ is a finitely generated projective generator in the category of right $E$-modules and the subrings $L \subseteq \text{End}(P,{+})$ such that $L$ is a submodule of the left $P$-module $\text{End}(P,{+})$ and $P$ is a finitely generated projective generator in the category of left $L$-modules.

References

[a1] N. Jacobson, "Galois theory of purely inseparable fields of exponent one" Amer. J. Math. , 66 (1944) pp. 645–648 DOI 10.2307/2371772 Zbl 0063.03019
How to Cite This Entry:
Jacobson-Bourbaki theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobson-Bourbaki_theorem&oldid=22607
This article was adapted from an original article by F. Kreimer (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article