3. Let \( \alpha = \sqrt{1 + \sqrt{3}} \), \( \beta = \sqrt{1 - \sqrt{3}} \), \( K_1 = \mathbb{Q}(\alpha) \), \( K_2 = \mathbb{Q}(\beta) \), \( K = \mathbb{Q}(\alpha, \beta) \).
1. Prove that \( K_1 \neq K_2 \), \( K_1 \cap K_2 = \mathbb{Q}(\sqrt{3}) = F \).
2. Prove that \( K_1/F \), \( K_2/F \) are Galois and find \( Gal(K_i/F) \) up to isomorphism.
3. Prove that \( K/F \) is Galois and find \( |Gal(K/F)| \).
4. Compute the elements of \( Gal(K/F) \) explicitly and prove \( Gal(K/F) \) is not cyclic.
Identify \( Gal(K/F) \) as a familiar group up to an isomorphism. Hint: one way is to
find \( \mathbb{Q}(\alpha, \beta) = \mathbb{Q}(\alpha, \text{something else}) \).
5. Prove \( K/\mathbb{Q} \) is Galois and find \( |Gal(K/\mathbb{Q})| \).
6. * (some extra pts) Find \( Gal(K/\mathbb{Q}) \). You do not have to solve this item!
