I'm reading class field theory by Neukirch, and i have a question seems simple but I can't find an answer by myself.
Now L/K is a cyclic Galois extension of number fields of degree p, and p is a prime number. In the page 130 of this book, it says that a prime of K that is not inert splits completely, i.e., decomposes into exactly p primes of L. But a prime of K which is not inert can be totally ramified, i.e., the index of ramification e=p, I don't know why this case hasn't been considered.
Thanks for your help!