Let $K$ be a number field.
For arbitrarily fixed positive integer $n$, fix $p_1,p_2,\ldots,p_n$ be a prime elements of ring of integers of $K$.
Does there exists a quadratic extension $L$ of $K$, in which $p_1,p_2,\ldots,p_n$ ramifies in $L/K$ ?
My try: From Dedekind's Discriminant theorem, we need to find $L/K$: quadratic such that all $p_i$ divides relative discriminant $\Delta_{L/K}$.
In the case $K=\Bbb{Q}$, Let $D=p_1p_2\cdots p_n$, then, $\Delta_{L/K}$ is $D$ or $4D$, thus $L=K(D)$ is enough in this case.
But in the case $K$ is not $\Bbb{Q}$, I'm stucking with how to find $L/K$: quadratic such that all $p_i$ divides $\Delta(L/K)$ because in this case, $\Delta{L/K}$ is not just $p_1・・・p_n$. I'm stucking with calculating $\Delta(L/K)$.
If there are any book or pdf dealing with this proposition, I would also welcome any references. Thank you in advance.