# Math Help - Elements of Extension Fields

1. ## Elements of Extension Fields

Prove that $\sqrt{7}$ is not an element of $\mathbb{Q}(\sqrt{3 + \sqrt{2}}).$

2. Originally Posted by h2osprey
Prove that $\sqrt{7}$ is not an element of $\mathbb{Q}(\sqrt{3 + \sqrt{2}}).$
here is one way to solve the problem:
let $\sqrt{3+\sqrt{2}}=a$ and $\sqrt{3-\sqrt{2}}=b$. if $\sqrt{7} \in \mathbb{Q}(a)$, then $b \in \mathbb{Q}(a)$ because $ab = \sqrt{7}$. that means $\mathbb{Q}(a)$ is the splitting field of $x^4-6x^2+7$. thus $\mathbb{Q}(a)/\mathbb{Q}$ is Galois and hence $|\text{Gal}(\mathbb{Q}(a)/\mathbb{Q})|=[\mathbb{Q}(a):\mathbb{Q}]=4.$ this is a contradiction because the galois group of $x^4-6x^2+7$ is the dihedral group of order $8$.