On some 2-extension of the field $\mathbb Q$ of rational numbers

It is proved that the field $\mathbb Q$ of rational numbers has one and only one normal 2-extension $\mathbb Q_{(2,\infty)}/\mathbb Q$ with the groupe isomorphic to $Z_2*\mathbb Z/2$. If $\Omega$ the maximal subfield of a real-closed field not contain in $\sqrt 2$, then the algebraic closure $\overline\Omega$ is isomorphic to the field $\Omega\underset{\mathbb Q}{\otimes}\mathbb Q_{(2,\infty)}$.