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Oddly Asymmetric
Oddly Asymmetric
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Promoted from stack.exchange since I received no response:
Jacobson developed a 'Galois' correspondence for purely inseperable extensions of exponent 1 (only consisting of pth roots) $K/k$, where he showed there was a correspondence between its subfields $k\subset{F}\subset{K}$ and finite restricted Lie sub-algebras of $k$-derivations given below: $$F\mapsto \mathcal{D}_F(K)$$ $$\mathcal{D'}\mapsto F, \text{the field of constants of} \space \mathcal{D'}$$ Davis extends this to exponent 2 extensions, and conjectures that higher derivation Lie-algebras should correspond to higher exponents.
Firstly, I'd like to ask for a list of books or articles that explore these ideas, especially the research done in 1968-1981, or more recently. I found the original papers in the American Mathematical Society very inaccessible, and have mostly been working off Lindsay Childs' survey of this research, but would like to learn more.

The other question I have, is if this theory is applicable, or can be extended, to the perfect closure of a field (in its algebraic closure, or more generally): is there a corresponding notion of an 'absolute lie-algebra of derivations' to that of the absolute galois group? If so please provide me with references to this too.

Promoted from stack.exchange since I received no response:
Jacobson developed a 'Galois' correspondence for purely inseperable extensions of exponent 1 (only consisting of pth roots) $K/k$, where he showed there was a correspondence between its subfields $k\subset{F}\subset{K}$ and finite restricted Lie sub-algebras of $k$-derivations given below: $$F\mapsto \mathcal{D}_F(K)$$ $$\mathcal{D'}\mapsto F, \text{the field of constants of} \space \mathcal{D'}$$ Davis extends this to exponent 2 extensions, and conjectures that higher derivation Lie-algebras should correspond to higher exponents.
Firstly, I'd like to ask for a list of books or articles that explore these ideas, especially the research done in 1968-1981, or more recently. I found the original papers in the American Mathematical Society very inaccessible, and have mostly been working off Lindsay Childs' survey of this research, but would like to learn more.

The other question I have, is if this theory is applicable to the perfect closure of a field (in its algebraic closure, or more generally): is there a corresponding notion of an 'absolute lie-algebra of derivations' to that of the absolute galois group? If so please provide me with references to this too.

Promoted from stack.exchange since I received no response:
Jacobson developed a 'Galois' correspondence for purely inseperable extensions of exponent 1 (only consisting of pth roots) $K/k$, where he showed there was a correspondence between its subfields $k\subset{F}\subset{K}$ and finite restricted Lie sub-algebras of $k$-derivations given below: $$F\mapsto \mathcal{D}_F(K)$$ $$\mathcal{D'}\mapsto F, \text{the field of constants of} \space \mathcal{D'}$$ Davis extends this to exponent 2 extensions, and conjectures that higher derivation Lie-algebras should correspond to higher exponents.
Firstly, I'd like to ask for a list of books or articles that explore these ideas, especially the research done in 1968-1981, or more recently. I found the original papers in the American Mathematical Society very inaccessible, and have mostly been working off Lindsay Childs' survey of this research, but would like to learn more.

The other question I have, is if this theory is applicable, or can be extended, to the perfect closure of a field (in its algebraic closure, or more generally): is there a corresponding notion of an 'absolute lie-algebra of derivations' to that of the absolute galois group? If so please provide me with references to this too.

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Oddly Asymmetric
Oddly Asymmetric
  • 295
  • 1
  • 8

Jacobson-style Galois theory on perfect closure

Promoted from stack.exchange since I received no response:
Jacobson developed a 'Galois' correspondence for purely inseperable extensions of exponent 1 (only consisting of pth roots) $K/k$, where he showed there was a correspondence between its subfields $k\subset{F}\subset{K}$ and finite restricted Lie sub-algebras of $k$-derivations given below: $$F\mapsto \mathcal{D}_F(K)$$ $$\mathcal{D'}\mapsto F, \text{the field of constants of} \space \mathcal{D'}$$ Davis extends this to exponent 2 extensions, and conjectures that higher derivation Lie-algebras should correspond to higher exponents.
Firstly, I'd like to ask for a list of books or articles that explore these ideas, especially the research done in 1968-1981, or more recently. I found the original papers in the American Mathematical Society very inaccessible, and have mostly been working off Lindsay Childs' survey of this research, but would like to learn more.

The other question I have, is if this theory is applicable to the perfect closure of a field (in its algebraic closure, or more generally): is there a corresponding notion of an 'absolute lie-algebra of derivations' to that of the absolute galois group? If so please provide me with references to this too.