Discrete valuation

Definition

A discrete valuation is a function defined on the field of fractions of a discrete valuation ring that assigns to each non-zero element a non-negative integer, representing the 'order' of that element. This concept helps in understanding the structure of local fields and plays a crucial role in higher ramification theory, which studies how primes split in extensions of these fields and how their valuations behave.

5 Must Know Facts For Your Next Test

1. Discrete valuations are key in defining local fields, allowing us to work with fields that have good topological properties.
2. The valuation associated with a discrete valuation can be used to study the behavior of prime ideals in number fields and their extensions.
3. In ramification groups, discrete valuations help to classify the way primes split or remain inert when moving from a base field to an extension.
4. Higher ramification theory involves studying not just the first-level behavior of valuations but also their higher-level structures through ramification groups.
5. A discrete valuation leads to an associated valuation ring, which consists of all elements whose valuation is non-negative, making it useful for local analysis.

Review Questions

• How does a discrete valuation function contribute to the understanding of local fields?
  • A discrete valuation function provides a way to measure the 'size' or 'order' of elements in a local field, which is crucial for understanding its topology and algebraic structure. Since local fields are complete with respect to this valuation, it helps in analyzing convergence and limits within the field. This structure simplifies many problems in algebraic number theory and provides insight into the behavior of prime ideals within these fields.
• Discuss the role of discrete valuations in ramification theory and how they affect prime splitting in extensions.
  • In ramification theory, discrete valuations allow mathematicians to classify how primes behave when extending fields. They help determine whether a prime remains inert, splits completely, or splits into several factors, thus providing information on how many times the prime appears in the factorization of ideals. The concepts of ramification groups are also built upon these valuations, creating layers that describe the fine structure of how primes act in extensions.
• Evaluate the importance of higher ramification groups in relation to discrete valuations and their applications in number theory.
  • Higher ramification groups build on the foundation laid by discrete valuations by examining more complex behaviors as we move through successive extensions. They provide deeper insights into how primes split and interact at various levels, revealing intricate patterns that are essential for advanced number theoretic applications. Understanding these groups can lead to significant results in arithmetic geometry and the study of Galois representations, demonstrating the critical role that discrete valuations play in modern algebraic number theory.

Related terms

Valuation ring: A valuation ring is a special type of integral domain that allows us to measure the size of its elements through a discrete valuation.

Ramification theory: Ramification theory studies how primes in a number field extend to primes in its field extensions, particularly looking at how they behave in terms of valuation.

Local field: A local field is a complete field with respect to a discrete valuation, typically arising as a completion of a number field or function field.