# 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 $\mathcal{O}$ be the ring of integers of a number field.  If $p$ is a rational prime number, then the principal ideal $(p)$ of $\mathcal{O}$ does not coincide with  $(1)=\mathcal{O}$  and thus $(p)$ has a set of prime ideals of $\mathcal{O}$ 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 $\mathcal{O}$ 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.

Title number of prime ideals in a number field NumberOfPrimeIdealsInANumberField 2013-03-22 19:12:51 2013-03-22 19:12:51 pahio (2872) pahio (2872) 4 pahio (2872) Theorem msc 11R04