number of prime ideals in a number field

Theorem.  The ring of integers of an algebraic number field contains infinitely many prime ideals.

Proof.  Let $𝒪$ be the ring of integers of a number field.  If $p$ is a rational prime number, then the principal ideal $(p)$ of $𝒪$ does not coincide with  $(1)=𝒪$  and thus $(p)$ has a set of prime ideals of $𝒪$ as factors.  Two different (positive) rational primes $p$ and $q$ satisfy

$$\gcd ((p),(q))=(p,q)=(1),$$

since there exist integers $x$ and $y$ such that  $xp+yq=1$  and consequently  $1\in (p,q)$.  Therefore, the principal ideals $(p)$ and $(q)$ of $𝒪$ have no common prime ideal factors.  Because there are http://planetmath.org/node/3036infinitely many rational prime numbers, also the corresponding principal ideals have infinitely many different prime ideal factors.