I tried to show that $\mathbb{Q}(\sqrt{p}\: |\: p\text{ prime})$ is a normal extension of $\mathbb{Q}$.
For finite extension we have that normal Extensions are splitting fields and vice versa, but how can I argue in the case of an infinite extension ?
Is there an argument that the infinite union of normal extensions is normal (If yes, how can I proof that?) or is there a simple argument for that example?
Any $\Bbb{Q}$-algebra homomorphism from $\mathbb{Q}(\sqrt{p}\: |\: p\text{ prime})$ to its algebraic closure sends $\sqrt{p}$ to $\pm\sqrt{p}$, whence it sends $\mathbb{Q}(\sqrt{p}\: |\: p\text{ prime})$ to itself which is normal over $\Bbb{Q}$.
Equivalently it is the splitting field of an infinite set of polynomials $x^2-p\ | \ p$ prime.