An infinite extension of $\mathbb Q$

Question

Let $S=\{\sqrt p \in \mathbb R | p $ is a primer number$\}$.

How can I show that $\mathbb Q(S)|\mathbb Q$ is an infinite field extension?

Answer 1

It follows directly from the infinity of primes and the fact that $\left[\mathbb Q[\sqrt p_1,\dots,\sqrt p_n]:\mathbb Q\right]=2^n$ for $p_1,\dots,p_n$ distinct prime numbers. This last fact is the theme of question 113689.