Understand the \$ p \$ part of the discriminant of a totally real number field with a prime number greater than \$ p \$

Let $$K$$ to be a totally real number field of Galois, and suppose that there is only one prime number above $$p$$, with branching index $$leq p-1$$. Yes $$K_p$$ is the completion of $$K$$ at the highest $$p$$, the claim is that the $$p$$-part of the discriminant of $$K$$ is equal to the discriminant of $$K_p$$.

I came across this information by reading Washington's "Introduction to Cyclotomic Fields," where he mentions in the proof of proposition 5.33 that "the $$p$$-part of the discriminant of $$K$$ is equal to the discriminant of $$K_p$$& # 39 ;, where the configuration is as described above. It's not clear to me how the basics for $$mathcal O_K$$ and $$mathcal O_ {K_p}$$ are related, so I'm not sure how to make sense of it. I've tried to decompress it a little bit by looking at the example where $$K = mathbb Q ( zeta_p) ^ +$$ is the maximum real subfield of $$mathbb Q ( zeta_p)$$, in which case $$(p) = (1- zeta_p) ^ {p-1}$$ is totally branched in $$mathbb Q ( zeta_p)$$, so $$K$$ satisfies the hypotheses. But even in this example, I have trouble calculating the relevant discriminants, let alone understanding this in general …