Disclaimer - cross-posting: I already posted this question on MSE, here. In line with the accepted answer of this meta question, I am also asking it here, since it is a research-level question and it has been seven months since I posted the MSE question, with no comments or answers.

Let $L/K$ be a Galois extension of $p$-adic fields with Galois group $V_4 = C_2 \times C_2$, and write $d_{L/K}$ for its discriminant, which is an ideal of $\mathcal{O}_K$. The extension $L/K$ has three quadratic intermediate fields $E_1,E_2,E_3$. I would like a reference for the fact that $$ d_{L/K} = d_{E_1/K}\cdot d_{E_2/K}\cdot d_{E_3/K}. $$ This is true because of the conductor-discriminant formula; we know that $$ d_{L/K} = \prod_{\chi} f(\chi)^{\chi(1)}, $$ where $\chi$ ranges over the irreducible complex characters of $V_4$, and $f(\chi)$ is the conductor of this character. The trivial character has conductor $1$. The three nontrivial characters $\chi_1,\chi_2,\chi_3$ correspond to the extensions $E_1/K, E_2/K, E_3/K$, and $f(\chi_i) = d_{E_i/K}$, again by the conductor-discriminant formula. The result follows.

This argument is quite long-winded, and referencing it properly would make it even longer. Can anyone give me a reference for the fact?