Trouble with two Galois theory questions

[LARaiders]

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)

 

[Siron]

1. What you have to do is to find the minimal polynomial of [tex]\sqrt{7}[/tex] and [tex]\sqrt{5}[/tex], that wil give you the degree [tex][\mathbb{Q}(\sqrt{5},\sqrt{7}):\mathbb{Q}][/tex].
Remark, note that
[tex][\mathbb{Q}(\sqrt{5},\sqrt{7}):\mathbb{Q}]=[\mathbb{Q}(\sqrt{5},\sqrt{7}),\mathbb{Q}(\sqrt{7})][\mathbb{Q}(\sqrt{7}):\mathbb{Q}][/tex]

 

[TheBigBadBen]

1)
Try

\(\displaystyle f(x) = (x^2-7)(x^2-5)\)

2)

What does

\(\displaystyle \phi \in Gal(\mathbb{Q}(4 \sqrt{3})|\mathbb{Q})\) do to the roots of the minimal polynomial?