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in this lecture we link degree of extension with ramification so let us first fix the terminology so let a be a dedicar domain with field of fractions k and let P be the integral closure of a and this B lies in a finite separable extension l over k so you take this primary LP in a and then you see that this splits in be like this and you can say fi is a degree of extension p over p I and so this is the degree of extension fi so first lemma so this term has kind of tautological so prime ideal P divides B so P divides p essentially means that this capital P occurs in the factorization of this smaller p in green so P divides me p this means this statement right here so this is true if and only if if you contract this bigger prime ideal back to a so this bigger prime area lies in be if you contracted back to a you will just get this p so first implication in this direction so that is this capital P divides this small p so now p would lie in P intersection a now this is obvious because this