Number of Primes in Ring of Integers of a Number Field

Solution 1:

The answer is yes. One somewhat high-powered way to see this is as follows:

Any class in the ideal class group contains infinitely many prime ideals. (This is part of the generalization of Dirichlet's theorem on primes in arith. progressions to general number fields.)

In particular, the trivial class does. That is, there are infinitely many principal prime ideals, and the generators of these ideals are then the desired infinitely many prime elements.

I'm not that there is a simpler approach then this one, since the question really is very analogous to Dirichlet's theorem, which requires $L$-function machinery to prove. (And that is what is used to prove the result I am quoting too.)

Solution 2:

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.