Reference request for a basic result on relative differents & discriminants

I am looking for a better reference for the results in this extremely short and elementary paper:

Tôyama, Hiraku, `A note on the different of the composed field', Kōdai Math. Sem. Rep. 7 (1955), 43–44.

In case the link stops working, the primary theorem states that if $K_1$ and $K_2$ are finite extensions of a number field $K_0$ with compositum $K_3 = K_1K_2$, then $$\text{lcm}\left(\mathcal{D}_{K_1/K_0},\mathcal{D}_{K_2/K_0}\right) ~\big|~ \mathcal{D}_{K_3/K_0} ~\big|~ \mathcal{D}_{K_1/K_0}\mathcal{D}_{K_2/K_0},$$ where $\mathcal{D}_{K_i/K_j}$ denotes the relative different.

As a corollary, if the relative discriminants $D_{K_1/K_0}$ and $D_{K_2/K_0}$ are relatively prime, then $$D_{K_3/K_0} = D_{K_1/K_0}^{[K_2:K_0]} \cdot D_{K_2/K_0}^{[K_1:K_0]}.$$

It's a very elementary result, and I can find the statement of at least the corollary in several places, but only stated in the case where $K_0 = \mathbb{Q}$. The Tôyama paper is a little frustrating to read because it contains some small errors, plus the typesetting is offensive to modern eyes. I would expect this to be in a basic algebraic number theory book. Does anyone know a nicer reference?

Edit: I meant to assume that $K_1$ and $K_2$ are linearly disjoint over $K_0$, so $K_1 \cap K_2 = K_0$. (Algebraic closure fixed before-hand, etc.) I think maybe Tôyama was assuming this tacitly.

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