mathbbZp

The symbol mathbb{Z}_p represents the ring of p-adic integers, where p is a prime number. It is a fundamental object in number theory and abstract algebra. The p-adic integers can be thought of as an extension of the integers, analogous to how the real numbers extend the rational numbers.

Elements of mathbb{Z}_p can be represented as infinite sequences of digits in base p, of the form ..., a_2 a_1 a_0, where each a_i is an integer such that 0 <= a_i < p. Addition and multiplication in mathbb{Z}_p are performed similarly to how they are done with integers, but with carries propagating infinitely to the left.

The construction of mathbb{Z}_p often involves the concept of inverse limits. Specifically, mathbb{Z}_p is the inverse limit of the rings of integers modulo powers of p, i.e., mathbb{Z}_p = lim_{leftarrow} mathbb{Z}/p^n mathbb{Z}. This means that an integer x is in mathbb{Z}_p if and only if for every n, there exists an integer x_n in mathbb{Z}/p^n mathbb{Z} such that x_n is congruent to x modulo p^m for all m < n, and x_n is congruent to x_{n+1} modulo p^n.

The p-adic integers form a complete normed ring with respect to the p-adic absolute value. This norm allows for the development of a p-adic analysis, which has applications in number theory, algebraic geometry, and mathematical physics. For instance, Hensel's Lemma provides a way to solve polynomial equations in mathbb{Z}_p. The field of p-adic numbers, denoted mathbb{Q}_p, is the field of fractions of mathbb{Z}_p.