number of prime ideals in a number field

Theorem.  The ring of integersMathworldPlanetmath of an algebraic number fieldMathworldPlanetmath contains infinitely many prime idealsMathworldPlanetmathPlanetmath.

Proof.  Let 𝒪 be the ring of integers of a number field.  If p is a rational prime number, then the principal idealMathworldPlanetmath (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


since there exist integers x and y such that  xp+yq=1  and consequently  1(p,q).  Therefore, the principal ideals (p) and (q) of 𝒪 have no common prime ideal factors.  Because there are 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