Valuation map

Definition

A valuation map is a mathematical function that assigns to each element of a field or ring a value that represents its 'size' or 'magnitude' in a specific sense. This concept is fundamental in algebraic number theory, particularly in the study of discrete valuations and valuation rings, where it helps to classify elements based on their divisibility properties and to construct corresponding algebraic structures.

5 Must Know Facts For Your Next Test

1. Valuation maps can be defined for both local fields and global fields, serving as tools to study their arithmetic properties.
2. For any element in a field, the valuation map typically assigns values based on how many times that element can be divided by a given prime or unit.
3. The image of a valuation map is often the set of integers or non-negative integers, reflecting the concept of size or divisibility.
4. Every valuation ring associated with a valuation map contains all elements whose valuations are non-negative, establishing a clear link between the two concepts.
5. Valuation maps play a critical role in the theory of completions, as they help in understanding how fields behave under various algebraic operations.

Review Questions

• How does a valuation map influence the structure of its corresponding valuation ring?
  • A valuation map directly determines which elements belong to the valuation ring by assigning non-negative values to elements. This creates a structure where all elements that have non-negative valuations are included in the ring, thus enforcing divisibility conditions. Consequently, the properties of the valuation map shape the entire framework of the valuation ring, dictating how elements interact under addition and multiplication.
• In what ways do valuation maps facilitate the understanding of local fields in algebraic number theory?
  • Valuation maps are essential for local fields as they provide a way to measure size and proximity among elements. By examining how these maps operate within local fields, we can better understand concepts like completion and how algebraic numbers behave under various operations. The insights gained from these maps help clarify the arithmetic structure of local fields and their applications in number theory.
• Evaluate the significance of valuation maps in classifying algebraic numbers and their properties.
  • Valuation maps are crucial for classifying algebraic numbers because they provide a systematic way to analyze divisibility and size. By assigning values based on how often an algebraic number can be divided by primes or units, these maps reveal deep insights into the structure of number fields. The classifications made possible by valuation maps enhance our understanding of algebraic properties such as ramification and local behavior, allowing for more comprehensive theories in algebraic number theory.

Related terms

Discrete valuation: A discrete valuation is a specific type of valuation that takes on only integer values and is associated with a valuation ring, representing how elements of the ring can be measured in terms of divisibility.

Valuation ring: A valuation ring is a special kind of integral domain that corresponds to a valuation, containing all elements with a non-negative valuation, thus allowing for a structured way to study the algebraic properties of numbers.

Local field: A local field is a field that is complete with respect to a discrete valuation and has a finite residue field, providing an important context for understanding the behavior of algebraic numbers and their valuations.