number of prime ideals in a number field
Proof. Let be the ring of integers of a number field. If is a rational prime number, then the principal ideal of does not coincide with and thus has a set of prime ideals of as factors. Two different (positive) rational primes and satisfy
since there exist integers and such that and consequently . Therefore, the principal ideals and of 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|
|Date of creation||2013-03-22 19:12:51|
|Last modified on||2013-03-22 19:12:51|
|Last modified by||pahio (2872)|