They are a key step for factoring polynomials over the integers or the rational numbers. It is thus isomorphic to the field of the integers modulo . Dedekind gave at least two proofs based on the arithmetic of the Gaussian integers. Note that formula 82 and formula 83 for all nonzero integers formula 84. Examples of algebraic integers are , and . The Euclidean algorithm for computing the greatest common divisor of two integers is one example. A permutation is a function that reorders this set of integers. Consider the set of all -digit binary integers. Many of the applications described above for integers carry over to polynomials. We denote the multiplicative subgroup of the modular integers by formula 40. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. More generally, the probability of "k" randomly chosen integers being coprime is formula 13. Unicode can be encoded in units of 8-bit, 16-bit, or 32-bit integers. The natural numbers form a subset of the integers. The factorial function is the product of all of the positive integers through .