Completion Map

Definition

A completion map is a function that takes a number field and produces its completion with respect to a specific valuation, typically resulting in a field that behaves well under certain algebraic operations. This process allows for the analysis of number fields in a more manageable setting, where issues of convergence and limits can be better understood. Completions help to study the properties of number fields and their extensions by providing a clearer view of local behavior at various places.

5 Must Know Facts For Your Next Test

1. The completion map transforms a number field into a complete space with respect to the chosen valuation, enabling the study of convergence and limits.
2. There are different types of completions based on whether you are working with p-adic or real valuations, leading to different structures such as local fields.
3. The completion of a number field with respect to an archimedean valuation results in the reals or complexes, while p-adic completions lead to p-adic fields.
4. Completions preserve many algebraic properties of the original number field, making them useful in local-global principles.
5. Understanding completion maps is crucial for studying integral elements and local rings in algebraic number theory.

Review Questions

• How does the completion map relate to the concept of valuation in number theory?
  • The completion map is closely tied to valuations as it relies on them to define how we measure 'size' within the number field. By selecting a specific valuation, we can apply the completion map to transform the number field into its completion. This process makes the structure more manageable and provides insights into local behavior. Valuations essentially guide us on which 'local' aspects are being examined through the completion.
• Discuss the implications of completing a number field with respect to different types of valuations.
  • Completing a number field with respect to different valuations leads to distinct results. For instance, when using an archimedean valuation, the completion yields real or complex numbers, giving insight into continuous behavior. In contrast, using p-adic valuations results in p-adic fields, which focus on discrete behavior. These different completions highlight various aspects of the number field, allowing mathematicians to use tools from both analysis and algebra in their studies.
• Evaluate the significance of completion maps in relation to the broader framework of algebraic number theory and its applications.
  • Completion maps hold significant importance within algebraic number theory as they facilitate a deeper understanding of number fields by allowing us to analyze local properties effectively. They serve as crucial tools for establishing connections between local and global properties through concepts like local-global principles. Additionally, they enable the study of integral elements and help construct objects like adèle rings that unify information across all completions. This interconnectedness enriches our comprehension of arithmetic geometry and aids in solving problems across various areas in mathematics.

Related terms

Valuation: A valuation is a function that assigns values to elements of a field, reflecting their size or multiplicative structure, and is key in defining completions.

Adèle Ring: The adèle ring is a construction in number theory that combines completions of a number field at all places, allowing for a unified approach to studying global properties.

Local Field:

A local field is a complete field with respect to a discrete valuation and is often obtained as a completion of a number field at a specific place.