Prime Ideal

In ring theory, a prime ideal is an ideal P of a commutative ring R such that if the product ab of two elements lies in P, then at least one of a or b lies in P. In the ring of integers O_K_ of an algebraic number field, prime ideals are the correct generalization of prime numbers: they factorize uniquely, they control ramification in extensions, and they are the local building blocks of the Dedekind zeta function. The quotient O_K_ / P is always a finite field, and the norm N(P) is its cardinality.

Prime ideals are not merely generalized primes. They are the geometric points of arithmetic schemes — the places where number theory becomes geometry. A number field without its prime ideals is like a manifold without its points: formally possible, structurally meaningless.