# 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

$$\mathrm{gcd}((p),(q))=(p,q)=(1),$$ |

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 |
---|---|

Canonical name | NumberOfPrimeIdealsInANumberField |

Date of creation | 2013-03-22 19:12:51 |

Last modified on | 2013-03-22 19:12:51 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 4 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 11R04 |