10.1 Galois Groups and Separability

1. First ε img gal(E : F) because ε(a) = a for all a img F. If σ, τ img gal(E : F) then σ(a) = a for all a img F, so a = σ−1(a) for all a; hence (since σ−1 is an automorphism) σ−1 img gal(E : F). Finally στ(a) = σ[τ(a)] = σ(a) = a, so στ img gal(E : F).
3. Let σ(ui) = τ(ui) for all i where σ, τ img gal(E : F). If img, write img, ai img F. Then

img

As was arbitrary, this shows that σ = τ.
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