L-adic numbers

Definition

l-adic numbers are a system of numbers used in number theory, which extends the concept of the usual integers to include a new way of measuring 'closeness' based on a prime number 'l'. They are constructed using the l-adic metric, allowing mathematicians to study properties of numbers and their relationships in a more refined way. This system is particularly useful in the study of l-adic representations and various aspects of algebraic geometry.

5 Must Know Facts For Your Next Test

1. l-adic numbers can be thought of as an alternative way of completing the rational numbers, similar to how real numbers are formed from rational numbers.
2. They allow for a different approach to solving Diophantine equations by considering solutions in terms of their l-adic properties.
3. l-adic representations relate to Galois groups, where these representations are often studied over l-adic fields rather than traditional fields.
4. In arithmetic geometry, l-adic numbers provide a framework for understanding points on varieties over finite fields and contribute to the study of rational points.
5. The topology induced by the l-adic metric makes l-adic numbers useful in understanding various concepts such as convergence and continuity within number theory.

Review Questions

• How do l-adic numbers extend our understanding of integers in number theory?
  • l-adic numbers extend our understanding of integers by introducing a new way to measure 'closeness' based on prime factors. Unlike traditional absolute values that measure distance from zero, l-adic valuation focuses on divisibility by a specific prime 'l'. This provides insight into properties of integers and rational numbers that can reveal hidden relationships and solutions, especially when dealing with Diophantine equations.
• Discuss the role of l-adic representations in the study of Galois groups and their implications in number theory.
  • l-adic representations play a crucial role in connecting Galois groups with various algebraic structures. By examining how these groups act on l-adic fields, mathematicians can uncover deep relationships between field extensions and their arithmetic properties. The insights gained from studying these representations can lead to breakthroughs in understanding solvability conditions for polynomial equations and provide tools for analyzing rational points on algebraic varieties.
• Evaluate the significance of l-adic numbers in arithmetic geometry and how they facilitate the study of rational points on varieties.
  • l-adic numbers are significant in arithmetic geometry as they provide a rigorous framework for analyzing rational points on varieties over finite fields. By incorporating l-adic techniques, researchers can utilize the rich structure of these numbers to investigate questions related to congruences, divisibility, and local-global principles. This approach has led to advancements in understanding how solutions behave under various conditions and has expanded the toolkit available for tackling complex problems within the discipline.

Related terms

l-adic valuation: The l-adic valuation is a function that assigns to each rational number a non-negative integer indicating how many times the prime l divides that number, thus giving a measure of its 'size' in the l-adic sense.

p-adic numbers:

p-adic numbers are a specific type of l-adic number where 'l' is replaced by a prime 'p', providing a complete field used extensively in number theory.

Galois representation:

A Galois representation is a homomorphism from the Galois group of a field extension to a linear group, which can often be studied using l-adic numbers.