Relative degree of a prime over a number field (notation from Algebraic Number Fields from Gerald J. Janusz) [closed]

Question

I´m working with "Algebraic Number Fields" from Gerald J. Janusz (1. edition from 1973) and I have a question about his notation.

In chapter IV proposition 4.5 he states if K is an algebraic number field and S is the set of primes of K which have relative degree one over Q then S is an infinte set.

Up to this point Janusz just defined the relative degree of a prime over R, with R as a ring of integers.

Can someone tell me, if there is a difference between the relative degree of a prime over a number field and the relative degree of a prime over a ring of integers?

Thanks in advance Julian

Answer 1

No, Janusz clearly refers to the relative degree over the ring of integers of $\mathbb Q$, that is $\mathbb Z$. It is very common in algebraic number theory to speak of something relative to a number field, when in reality it means relative to its ring of integers. For instance, we commonly talk of ideals of $K$, which should only mean $(0)$ and $K$, when we actually mean ideals of $\mathcal O_K$.