Extension of a valuation

Definition

An extension of a valuation is a way to enlarge a valuation defined on a field to a larger field while preserving its properties. This concept is crucial when working with discrete valuations, as it allows us to analyze how the valuation behaves when we move from one field to an extension field, influencing the study of valuation rings and their properties.

5 Must Know Facts For Your Next Test

1. The extension of a valuation can be unique, depending on the nature of the fields involved and the original valuation defined.
2. When extending valuations, it is important to consider whether the new field contains sufficient information about the original field's structure.
3. Extensions can reveal important information about algebraic and arithmetic properties, especially in local fields.
4. The process of extending a valuation often involves considering residue classes and their relationships within the valuation ring.
5. Understanding extensions of valuations is essential for studying completion processes in various algebraic settings.

Review Questions

• How does the extension of a valuation affect the structure of valuation rings?
  • The extension of a valuation plays a significant role in determining the structure of valuation rings because it can change how elements are classified regarding their divisibility by primes. When a valuation is extended to a larger field, the corresponding valuation ring may include additional elements that were not in the original ring. This change can lead to new insights into the properties of the larger field, particularly regarding its ideal structure and divisibility characteristics.
• Discuss the uniqueness conditions for extending valuations from one field to another and their implications.
  • The uniqueness of extending valuations depends on various factors, such as whether the extension field is algebraic or transcendental over the base field. In many cases, if the extension maintains certain properties, like being algebraically closed or complete, there may exist unique extensions that reflect these properties. These implications are critical in number theory and algebra since they can influence how certain equations behave when considered in larger fields, affecting solutions and other algebraic structures.
• Evaluate how understanding extensions of valuations contributes to broader concepts in algebraic number theory.
  • Understanding extensions of valuations contributes significantly to broader concepts in algebraic number theory by allowing mathematicians to analyze local-global principles and understand how different fields relate to one another. For example, by examining how valuations extend across fields, one can derive insights into factorization properties, Galois theory, and even Diophantine equations. This analysis deepens our understanding of arithmetic properties of numbers and facilitates connections between various areas in mathematics, including algebraic geometry and cryptography.

Related terms

Discrete valuation: A discrete valuation is a function that assigns to each non-zero element of a field a non-negative integer, capturing how 'divisible' that element is by a prime element.

Valuation ring: A valuation ring is a special type of integral domain associated with a valuation, consisting of all elements of the field that have non-negative valuation.

Hensel's lemma: Hensel's lemma provides conditions under which a polynomial equation has solutions in a p-adic field, and relates closely to the behavior of valuations in local fields.