Linguistic Premise Premise of Algebraic Linguistics 2-4 / September 22, 2007

Linguistic Premise

 Premise of Algebraic Linguistics 2-4

　　  TANAKA Akio

32 <automorphism group>

Algebraic system     X

Automorphism     bijectional homomorphism from X to X

All the automorphism has productive composition.      Automorphism group     Expression is Aut ( X )

33 <operate>

Ring     R

Automorphism group     Aut ( X )

Homomorphism of group     G → Aut( R )

G operates R.

Homomorphism is surjection.       G separates R faithfully.

34 <normal separable extension>

Finite group G operates field L faithfully.

Invariant subfield of L     k = L^G := { a ∈ L | σ( a ) = a ( ∀σ ∈ G )

Normal separable extension ( Galois extension )     [ L : k ] = | G |

35 <finite separable normal extension i.e. Galois extension>

Galois extension of L/k      G = Gal ( L/k )

Subfield of G     H

Invariant subfield of H     L^H = { a ∈ L | σ ( a ) = a  ∀ σ ∈ H }

Intermediate field of L/k     E    

H ( E )= { σ ∈ G | σ ( a ) = a  ∀ a ∈ E }   

36 <fundamental theorem of Galois theory>

Extension field is controlled by group theory.

Field of characteristic 0     k

Finite Galois extension     K ⊃ k

Intermediate field of K ⊃ k      M

Galois group     Gal ( K/k )

Subgroup of Gal ( K/k )     H

Galois correspondence

(1) M →^φ Gal ( K/M )

(2) H →^ψ L^H

Extension M ⊃ k  ⇔   Gal ( K/M ) is normal subfield of Gal ( K/k ).

Isomorphism     Gal ( M/k ) ≅ Gal ( K/k ) / Gal ( K/M )

Tokyo September 22, 2007

Sekinan Research Field of Language