# Trouble with two Galois theory questions

#### LARaiders

##### New member
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

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

$$\displaystyle f(x) = (x^2-7)(x^2-5)$$
$$\displaystyle \phi \in Gal(\mathbb{Q}(4 \sqrt{3})|\mathbb{Q})$$ do to the roots of the minimal polynomial?