**Disclaimer - cross-posting:** I already posted this question on MSE, [here][1]. In line with the accepted answer of [this meta question][2], 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? 


  [1]: https://math.stackexchange.com/questions/4703084/discriminant-of-a-v-4-extension-of-local-fields-is-the-product-of-discriminant
  [2]: https://meta.mathoverflow.net/questions/5012/can-i-ask-a-question-on-mathoverflow-and-also-on-another-site