Introduction to Abstract Algebra, Solutions Manual, 4th Edition

10.1 Galois Groups and Separability

1. First ε gal(E : F) because ε(a) = a for all a F. If σ, τ gal(E : F) then σ(a) = a for all a F, so a = σ⁻¹(a) for all a; hence (since σ⁻¹ is an automorphism) σ⁻¹ gal(E : F). Finally στ(a) = σ[τ(a)] = σ(a) = a, so στ gal(E : F).

3. Let σ(u_i) = τ(u_i) for all i where σ, τ gal(E : F). If , write , a_i F. Then

As was arbitrary, this shows that σ = τ.

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