An integer p is prime if it has exactly two positive divisorsMathworldPlanetmathPlanetmath. The first few positive prime numbersMathworldPlanetmath are 2,3,5,7,11,.

A prime number is often (but not always) required to be positive.

Prime numbers are very important objects in many fields of mathematics. The notion of prime number has been generalized in a number of ways.

In class field theory, one defines a prime to be a family of equivalent valuations; using this definition, the primes of the rational numbers are given by the positive prime numbers (for a prime p, the corresponding valuationMathworldPlanetmathPlanetmath is the p-adic absolute valueMathworldPlanetmathPlanetmath) and one extra prime usually denoted (corresponding to the usual absolute value on ).

In ring theory, one defines the notion of a prime idealMathworldPlanetmathPlanetmath, and also a notion of prime elementMathworldPlanetmath ( However, the notion of prime ideal is more natural in this context than the notion of prime element. For number fieldsMathworldPlanetmath, for example, one has unique factorizationMathworldPlanetmath of ideals into prime ideals but one does not always have unique factorization of elements into prime elements. The prime ideals of are the ideals generated by the prime numbers, as well as the zero idealMathworldPlanetmathPlanetmath. Note that the ideal generated by p is the same as the ideal generated by -p, so we can consider negative primes equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to positive primes from this viewpoint as well.

Title prime
Canonical name Prime
Date of creation 2013-03-22 11:51:52
Last modified on 2013-03-22 11:51:52
Owner djao (24)
Last modified by djao (24)
Numerical id 11
Author djao (24)
Entry type Definition
Classification msc 11A41
Classification msc 82-00
Classification msc 83-00
Classification msc 81-00
Classification msc 46L05
Synonym prime integer
Synonym prime number
Synonym rational prime
Related topic PrimeNumberTheorem