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 …