Jacobson-Bourbaki theorem

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