Ring of Integers

The ring of integers of an algebraic number field K, denoted O_K_, is the set of all elements of K that are roots of monic polynomials with coefficients in Z. It is the natural generalization of the ordinary integers to arbitrary number fields, and it serves as the stage on which all arithmetic in K is performed. O_K_ is always a Dedekind domain, and its ideal class group measures the failure of unique factorization of elements. The discriminant of K — an integer that encodes the ramification of primes in the extension — is one of the most powerful invariants in all of number theory.\n\nThe ring of integers is not merely a generalization of Z. It is the correction: Z is the ring of integers of Q, and Q is the least interesting number field. To start arithmetic with Z and then generalize is to learn geometry from a point.\n\n