Abstract Linear Algebra Question

[LARaiders12]

I am so confused with these two questions. Can anyone help me out?

1) Please find [Q((√7 , √5) : Q] by finding f(x) such that Q (√7 , √5) ≅ Q[x]/(f(x)),

2) Prove that φ(4root√3) = ± 4root√3, Given that φ ∈ Gal(Q(4root√3)|Q)

 

[daon2]

1. you need a minimal (irreducible) polynomial for sqrt(5) over Q. Then a minimal polynomial for sqrt(7) over Q(sqrt(5)). the degree of the extension will be the product of these because 5,7 are prime. or, you might want to note that Q(sqrt(5),sqrt(7)) = Q(sqrt(5)+sqrt(7))

2. the subgroups of the galois permutation group correspond to the subfields of Q(4root√3) containing Q. You must show that if there is a permutation f in G then it must only change the sign (there actuially isn;t much choice in the matter if you think about it.). The minimal polynomial is clearly x^2-48, of which Q(4sqrt(3))=Q(sqrt(3)) is the splitting field.

edit: actually i'm not sure if you meant \(\displaystyle \mathbb{Q}[\sqrt[4]{3}]\) or what you wrote \(\displaystyle \mathbb{Q}[4 \sqrt{3}]\).