2) Find Gal(F/Q) for the following fields F:
F = Q(ζ₅), where ζ₅ is a non-real fifth root of unity;
F = Q(⁵√3)
F = Q(∜2,i)
3) Is Z a group under subtraction? Prove your answer.
4) Let F be a field and let σ be an automorphism of F. Prove that the inverse function σ⁻¹ is also an automorphism of F.
5) Find orders of all elements in the cyclic group (Z/10Z, +). List all generators.
6) Find orders of all elements in the cyclic group (μ₈, •). List all generators.
7) Find orders of all elements in the group ((Z/20Z)ˣ, •). Is this a cyclic group?
8) Recall that Gal(Q(√3, √5)/Q) ≅ C₂ × C₂. What group is Gal(Q(√2, √3, √5)/Q) isomorphic to? Construct the isomorphism.
9) Prove that the groups (Z/6Z, +) and (μ₆, •) are isomorphic.
10) Show that Gal(Q(∜2, i)/Q) ≅ D₄.
11) Is it true that C₃ × C₃ is isomorphic to C₉? Is it true that C₃ × C₅ is isomorphic to C₁₅ ? Prove your answers.
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