Abstract Algebra Problems

1) Let α, β be two distinct roots of a cubic polynomial f ∈ Q[x] which is irreducible over Q. Moreover, assume that α ∈ R, but β ∉ R. Write ζ₃ for any non-real cube root of unity. Decide which of the following fields are equal, which are isomorphic (but not equal) and which are neither: Q(α), Q(β), Q(√-3), Q(√3, i), Q(ζ₃), Q(∜5), Q(i∜5), Q(-i,∜5). Include some justification.

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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