Confusion in definition of prime of a algebraic number field I am reading from Algebraic Number Theory by J. Neukirch on my own.
I am confused with the definition of prime divisors of a field extension.
This is on page page 44

Later in Chapter 3, he has defined it in some different way. 

Am I missing something ?  Are these definition equivalent? 
 A: From a linguistic perspective these are similar. This was a choice made by the developers of the theory because of (the natural generalization of) Ostrowski's theorem, namely that the absolute values of a number field, $K$, are all induced by prime ideals of $K$ and by the (non-equivalent) embeddings into $\Bbb C$. In the case of $\Bbb Q$ it is literally Ostrowski's theorem as each prime ideal is associated with a unique class of equivalent absolute values--the usual absolute value inherited from $\Bbb R$ is associated with the prime ideal $(0)$.
However, you're right these aren't literally the same thing, they are just connected (hence the linguistic similarity). If you look closely you'll see the first definition is for "prime ideals" and the second is for "primes."  While we abuse the language and use these things interchangeably in practice, they are not literally the same thing:

One is a set satisfying some properties, the other is an equivalence class of functions satisfying some other properties.

A: You have asked if "these definitions are equivalent", but I see only one definition in your snippets.
Prime ideals are a general concept in abstract algebra. They are defined for any (commutative) ring, of which the ring of integers is a special case.
The term defined in the second snippet, a prime (of a number field), is a separate concept that is a priori unrelated to the more general definition of a prime ideal.
However, as one may guess, both concepts are somehow generalizing the concept of an (ordinary) prime number $p=2,3,5,\ldots$ in $\mathbb Z$.
