Let $L$ be a Galois extension of $\mathbb{Q}$ and $M$ a finite extension of $\mathbb{Q}$. A Theorem of Bauer tells that $Spl_1(M)\subset Spl(L)$ up to a finite number of exceptions is equivalent to $M\subset L$. Here $Spl(L)$ denotes the rational primes which splits in $L$ and $Spl_1(M)$ denotes the rational primes such that there exists a prime of inertial degree $1$ above.

This result can be found in Cox, Primes of the form $x^2+ny^2$ (Prop 8.20) or Neukirch (p. 135) for example.

$\textbf{My question}:$ Suppose that $L\cap M =\mathbb{Q}$ with $L$ Galois, Bauer's result impliesthe existence of infinitely primes in $Spl_1(M)$ which are not splitting in $L$. Can we ask for something stronger, I mean a positive density of such primes (and even quantify that density using the information on Galois groups)?

To be less general, in my case I am interested when $L$ is a quadratic extension.

Thanks in advance!