Then in the next step I want to show that the kernel of $\varphi_{L/K}$ is exactly $P_K$. Because I know excatly which prime ideals maps to $1 \in Gal(L/K$, I can see that every product of principal prime ideals will be mapped to the $1$. Every Product of even many non-principal prime ideals will be mapped to the $1$, too. Because the class number of $K$ is $2$ every product of non-principal ideals must be a principal ideal. Can I conclude from this that every non-principal(resp. principal) Ideal is the product of some principal ideals and uneven(resp. even) man non-principal ideals?

Ah, now I see: $p\mathcal{O}_K =(x+y\sqrt{-5})(x-y\sqrt{-5}) = (x^2 +5y^2)$ and this is exactly the case iff $p \equiv 1,9\ mod\ 20$. And for $p \equiv 11,13,17,19\ mod\ 20$ $p\mathcal{O}_K$ remains prime and is principal. Now I only need to compute $\varphi(2\mathcal{O}_K)$ and $\varphi(5\mathcal{O}_K)$. Two another questions: Is $P_K$ generated by the principal prime ideals and how can I see in which ways $2\mathcal{O}_K,5\mathcal{O}_K$ splits in $L$? And thank you for your help.

When you talk about norm, do you mean $\mathcal{N}(\mathfrak{p}) = |\mathcal{O}_K/\mathfrak{p}|$ for a prime ideal $\mathfrak{p}$ in $K$?