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 = LG := { 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 LH = { 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 LH
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
No comments:
Post a Comment