Definition : Let $K$ be a field with $char(K)=0$ , let's define $ K^{quad}\subset \overline{K}$ by: $ K^{quad} = \bigcup\limits_{i \geqslant 1} {K_i } $ where $ K_1 = K $ $ K_{i + 1} = K_i(\sqrt{K_i}) $
and where $K_i(\sqrt{K_i})$ means the field generated over $K_i$ by all the roots of $K_i$ i.e elements in an algebraic closure of $K_i$ such that $x^2 \in K_i $
Prove that if $K\subset \Bbb C$ is such that it's closed under conjugation, then $K^{quad}$ is also closed under conjugation.