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Homenumber of prime ideals in a number field

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# 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 infinitely many rational prime numbers, also the corresponding principal ideals have infinitely many different prime ideal factors.

## Mathematics Subject Classification

11R04*no label found*

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