TITLE: Number of Primes in Ring of Integers of a Number Field
QUESTION [10 upvotes]: In the ring of integers of an algebraic number field, we say that an algebraic integer $p$ is prime if, whenever $p$ divides product of two algebraic integers, $p$ divides one of them. An algebraic integer is said to be irreducible if it can not be a product of two algebraic integers, which are not units. 
I came across the following theorem: (1) The ring of integers of any algebraic number field contains infinitely many irreducible elements. (2) In the ring of algebraic integers of an algebraic number field, every prime element is irreducible (but not converse)".
To prove (1), I tried to use (2), and arrived at the following question.
Does the ring of integers of an algebraic number field contains infinitely many primes? If yes, does it follows from the argument of Euclid on the infiniteness of primes in $\mathbb{Z}$?

REPLY [2 votes]: user160609 answers your first question, in what I think is the best possible way.
As for your other question: you cannot necessarily do a Euclid-style argument in a number ring, even to prove that there are infinitely many irreducible elements, because you cannot exclude the possibility that the product $p_1\dots p_n+1$ is a unit.