1) Let Î±, Î² be two distinct roots of a cubic polynomial f âˆˆ Q[x] which is irreducible over Q. Moreover, assume that Î± âˆˆ R, but Î² âˆ‰ R. Write Î¶â‚ƒ 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(Î±), Q(Î²), Q(âˆš-3), Q(âˆš3, i), Q(Î¶â‚ƒ), Q(âˆœ5), Q(iâˆœ5), Q(-i,âˆœ5). Include some justification.

2) Find Gal(F/Q) for the following fields F:
F = Q(Î¶â‚…), where Î¶â‚… is a non-real fifth root of unity;
F = Q(âµâˆš3)
F = Q(âˆœ2,i)

3) Is Z a group under subtraction? Prove your answer.

4) Let F be a field and let Ïƒ be an automorphism of F. Prove that the inverse function Ïƒâ»Â¹ 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 (Î¼â‚ˆ, â€¢). List all generators.

7) Find orders of all elements in the group ((Z/20Z)Ë£, â€¢). Is this a cyclic group?

8) Recall that Gal(Q(âˆš3, âˆš5)/Q) â‰… Câ‚‚ Ã— Câ‚‚. What group is Gal(Q(âˆš2, âˆš3, âˆš5)/Q) isomorphic to? Construct the isomorphism.

9) Prove that the groups (Z/6Z, +) and (Î¼â‚†, â€¢) are isomorphic.

10) Show that Gal(Q(âˆœ2, i)/Q) â‰… Dâ‚„.

11) Is it true that Câ‚ƒ Ã— Câ‚ƒ is isomorphic to Câ‚‰? Is it true that Câ‚ƒ Ã— Câ‚… is isomorphic to Câ‚â‚… ? Prove your answers.

1) Let Î±, Î² be two distinct roots of a cubic polynomial f âˆˆ Q[x...