In Lorenz's Galois Theory book, there's a problem :
Why $\sqrt{15} \notin \mathbb{Q(\eta_{15})}$, where $\eta_{15}$ is a $15$-th primitive root of unity ?
But My question is about what it's written in the hint :
Why $\sqrt{5} \notin \mathbb{Q(\eta_{5})}$ ?
Now, I see that $G(\mathbb{Q(\eta_{15})}/Q)\simeq U_{15}\simeq \mathbb{Z}_{2^2}\times\mathbb{Z}_{2}$ and hence has $7$ subgroups: $\{(0,0)\}$, $\mathbb{Z}_{2^2}\times\mathbb{Z}_{2}$, etc.
There's written in that problem that the corresponding subfields of the extension $\mathbb{Q}(\eta_{15})/\mathbb{Q}$ are : $\mathbb{Q}(\sqrt{-3})$, $\mathbb{Q}(\sqrt5)$, $\mathbb{Q}(\sqrt{-15})$, $\mathbb{Q}(\sqrt{-3},\sqrt{5})$, $\mathbb{Q}(\eta_5)$ and $\mathbb{Q}(\eta_5+{\eta_5}^{-1})$: It's not hard that these subfields are distinct and hence they're only subfields of $\mathbb{Q}(\eta_{15})/\mathbb{Q}$.
Here is the hint : "$\sqrt{-3}\in\mathbb{Q}(\eta_3)$ and $\sqrt{5}\in\mathbb{Q}(\eta_5)$", but I can't understand why $\sqrt5$ lies in $\mathbb{Q}(\eta_5)$ and by this assumption, I've no idea how to continue right now.