We have to show if $K =\mathbb{ Q}(\{ \sqrt q : q \text{ is a prime integer}\}) \subset R$. Then $[K : \mathbb{Q}]$ is not finite.
Hint: First show, for any positive integer $r$, if $p_1,p_2,...,p_r,q$ are distinct prime integers, then $\sqrt q \notin \mathbb{Q}(\sqrt p_1,\sqrt p_2,...,\sqrt p_r)$ by induction. Consequently, we get a strictly infinite ascending chain of fields such that $\mathbb{Q} \subset \mathbb{Q}(\sqrt 2) \subset\mathbb{ Q}(\sqrt 2,\sqrt 3) \subset \cdots$ This shows that [K : Q] is not finite. And also clearly $[\mathbb{R}:\mathbb{Q}]$ is not finite
I think I have done the induction part.
Our induction hypothesis is: let $p_1,p_2,...,p_n$ be distinct prime integers (each is different from $q$) such that $\sqrt q \notin \mathbb{Q}(\sqrt p_1,\sqrt p_2,...,\sqrt p_n)$
As $q$ is a prime integer, $ \sqrt q \notin \mathbb{Q}$,the induction hypothesis is true for $n = 0$.let's assume the hypothesis for $n$. Now suppose $\sqrt q \in \mathbb{Q}(\sqrt p_1,\sqrt p_2,...,\sqrt p_n,\sqrt p_{n+1})$.
If $\sqrt q \in \mathbb{Q}(\sqrt p_1,\sqrt p_2,...,\sqrt p_n,\sqrt p_{n+1})$ then there exists $x,y \in \mathbb{Q}(\sqrt p_1, \sqrt p_2,...,\sqrt p_n)$ such that $\sqrt q=x+y \sqrt p_{n+1}$. If $y=0$ then $\sqrt q=x \in \mathbb{Q}(\sqrt p_1, \sqrt p_2,...,\sqrt p_n)$ which contradicts the induction hypothesis so $y \neq 0$. If $x=0$, then $\sqrt q =y \sqrt p_{n+1} \implies q=y^2 p_{n+1}$ also contradiction since $q$ and $p_{n+1}$ are distinct primes so $x \neq 0$. Now for $x \neq 0$ and $y \neq 0$ , $\sqrt q=x+y \sqrt p_{n+1} \implies q=x^2+2xy \sqrt p_{n+1} +p_{n+1} y^2 \implies \sqrt p_{n+1}=\frac{q-x^2-p_{n+1} y^2}{2xy} \in\mathbb{Q}(\sqrt p_1, \sqrt p_2,...,\sqrt p_n)$
now As $\sqrt p_{n+1} \in\mathbb{Q}(\sqrt p_1, \sqrt p_2,...,\sqrt p_n), so \sqrt q =x+y \sqrt p_{n+1} \in \mathbb{Q}(\sqrt p_1, \sqrt p_2,...,\sqrt p_n)$ which is again a contradiction.
So by induction hypothesis, for any positive integer $r$, if $p_1,p_2,...,p_r,q$ are distinct prime integers, then $\sqrt q \notin \mathbb{Q}(\sqrt p_1,\sqrt p_2,...,\sqrt p_r)$.
But I can't follow the other hints. How it's consequently follow the strictly infinite ascending chain of fields such that$ \mathbb{Q} \subset \mathbb{Q}(\sqrt 2) \subset\mathbb{ Q}(\sqrt 2,\sqrt 3) \subset\cdots$. And form this how to conclude $[K:\mathbb{Q}]$ is not finite. And how it's clearly follow that $[\mathbb{R}:\mathbb{Q}]$ is not finite.
