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 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