Proving that $\mathbb{Q}$ adjoin the square root of every prime is an infinite extension

Question

How would one show that $[\mathbb{Q}(\sqrt2, \sqrt3,...,\sqrt{p_n},...)]=\infty$? I know that we want to show there is no finite basis over the rationals, but I'm not sure how one would determine that such a basis does not exist.

Answer 1

Kummer theory tells us that as $ 2$, $3,\ldots,{p_n}$ generate a subgroup of order $2^n$ in $\Bbb Q^*/(\Bbb Q^*)^2$ then the extension field $K_n=\Bbb Q(\sqrt2,\ldots,\sqrt {p_n})$ has degree $2^n$ over $\Bbb Q$. So your field contains subfields of arbitrarily large degree over $\Bbb Q$.