Completions

Definition

Completions refer to a method of constructing a new mathematical object from an existing one by adding 'limits' or 'points' to make it more complete in terms of certain properties, like convergence. This concept is particularly important when examining local properties of mathematical structures, allowing us to connect these local perspectives with global characteristics, and is essential in the study of algebraic number theory.

5 Must Know Facts For Your Next Test

1. Completions allow mathematicians to work with more manageable and well-behaved structures, particularly in the context of number theory.
2. In algebraic number theory, completions help in analyzing the behavior of numbers and equations at various local places, like p-adic and real completions.
3. The process of completing a field can be thought of as filling in 'gaps' to ensure that limits and Cauchy sequences behave well within the structure.
4. The completion of the rational numbers with respect to the usual absolute value leads to the real numbers, while the p-adic completion yields p-adic numbers.
5. A local-global principle states that if a solution exists locally (in all completions), it might imply the existence of a solution globally in the original structure.

Review Questions

• How do completions help bridge the gap between local properties and global characteristics in algebraic structures?
  • Completions enable mathematicians to focus on local properties by constructing fields where sequences converge properly. By studying these completed fields, one can draw conclusions about solutions or behaviors in the original global structure. This approach allows for a clearer understanding of how local phenomena can reflect or influence the global scenario, particularly useful in problems involving Diophantine equations.
• Discuss how the completion of a field changes its properties and why this is significant for solving equations.
  • Completing a field alters its structure by including limits and making it more 'complete' concerning convergence. For instance, when completing the rational numbers, we gain access to real numbers which include limits of Cauchy sequences. This completeness is significant for solving equations because it allows for a better understanding of roots and behaviors at various places, leading to potential solutions that might not be visible in the original field.
• Evaluate the impact of completions on the local-global principle and how they influence solutions to algebraic equations.
  • Completions significantly influence the local-global principle by establishing connections between local solutions (found within completed fields) and potential global solutions. If an algebraic equation has solutions in every local completion, it suggests that there may also be a corresponding solution in the original field. This powerful relationship underscores the importance of completions in algebraic number theory, as it provides a pathway for verifying global existence by examining localized cases.

Related terms

Valuation: A function that assigns a value to elements of a field, measuring their 'size' or 'complexity', which is crucial for understanding completions.

Local Fields: Fields that are complete with respect to a valuation, often used in number theory to analyze local-global principles.

Henselian Field: A type of field that satisfies certain conditions related to the completion process, allowing for better handling of polynomial equations.