P-adic integers

Definition

P-adic integers are elements of the ring of p-adic integers, denoted as $$\mathbb{Z}_p$$, which is a system of numbers that extends the ordinary integers to include a form of 'infinitesimal' behavior with respect to a prime number p. This structure allows for a unique way of measuring distances between numbers based on divisibility by powers of p, making it useful in number theory and algebraic geometry.

5 Must Know Facts For Your Next Test

1. P-adic integers can be represented as infinite series in the form $$a_0 + a_1 p + a_2 p^2 + \ldots$$ where each $$a_i$$ is an integer between 0 and p-1.
2. They form a compact space under the p-adic topology, meaning every sequence of p-adic integers has a convergent subsequence.
3. P-adic integers are closed under addition and multiplication, making them a ring with unity.
4. The set of p-adic integers contains all ordinary integers, but not all p-adic numbers are integers; some may have 'infinite' negative powers of p.
5. The relationship between p-adic integers and algebraic K-theory is significant, as they help understand local-global principles and can play a role in the formulation of conjectures like Quillen-Lichtenbaum.

Review Questions

• How do p-adic integers relate to the concept of completion in number theory?
  • P-adic integers arise from completing the rational numbers with respect to the p-adic norm. This completion process creates a new structure that encapsulates different properties compared to real numbers. The completeness ensures that every Cauchy sequence of rational numbers converges to a p-adic integer, allowing for unique manipulations and solutions within this framework.
• Discuss the significance of p-adic integers in the context of local fields and their applications in algebraic K-theory.
  • P-adic integers are integral to understanding local fields, as they provide a means to study properties at a prime level. In algebraic K-theory, these integers help analyze structures that emerge from local-global phenomena. Their use in local fields aids in formulating and proving conjectures like Quillen-Lichtenbaum, which connects homotopy theory and number theory through p-adic methods.
• Evaluate how the properties of p-adic integers might influence the validity of the Quillen-Lichtenbaum conjecture within algebraic K-theory.
  • The validity of the Quillen-Lichtenbaum conjecture relies heavily on understanding how local phenomena, captured by structures like p-adic integers, can provide insights into global properties in algebraic K-theory. Since p-adic integers retain rich arithmetic properties and enable deep connections with cohomological aspects, their behavior can be crucial for proving or disproving conjectures that involve relations between K-theory and étale cohomology at various primes. This interplay reflects the centrality of p-adics in modern number theory and algebra.

Related terms

p-adic norm: A way to measure the size or distance between p-adic integers, which captures how divisible a number is by a power of the prime p.

p-adic completion:

The process of completing the rational numbers with respect to the p-adic norm, leading to the formation of p-adic numbers, which include both finite and infinite expansions.

Local fields: Fields that are complete with respect to a discrete valuation, such as the field of p-adic numbers, which plays a crucial role in local number theory.