Valuation ring

Definition

A valuation ring is a special type of integral domain that is associated with a valuation, which provides a way to measure the size of elements in a field. Specifically, it is a subring of a field where for any element in the field, either that element or its inverse (if it exists) belongs to the ring. This concept is crucial in understanding the structure of local fields and plays a key role in ramification theory, where it helps in analyzing how prime ideals behave under field extensions.

5 Must Know Facts For Your Next Test

1. Valuation rings are integral domains where for any non-zero element, either it or its inverse is contained within the ring.
2. Every discrete valuation gives rise to a unique valuation ring, which helps classify the properties of the field associated with the valuation.
3. Valuation rings can be used to study the behavior of primes in extension fields, especially in understanding how they split or ramify.
4. In a valuation ring, the set of non-units forms an ideal which is linked to the valuation itself, providing important structural insights.
5. The concept of maximal ideals in valuation rings relates directly to the valuation, as they correspond to prime elements with certain size properties.

Review Questions

• How does the structure of a valuation ring facilitate the study of discrete valuations and their properties?
  • The structure of a valuation ring is closely tied to discrete valuations because it allows for the classification of elements based on their size relative to a prime. Since every discrete valuation corresponds to its unique valuation ring, analyzing these rings helps us understand how different elements relate under valuation. By focusing on whether an element or its inverse belongs to the ring, we can better explore their behavior in extensions and how they interact with ideals.
• Discuss the role of valuation rings in local fields and how they contribute to our understanding of these fields.
  • Valuation rings play a crucial role in local fields because they provide the foundational structure necessary for understanding completions with respect to discrete valuations. Since local fields are complete and have finite residue fields, their associated valuation rings help determine properties such as uniformizers and maximal ideals. This understanding is essential when examining extensions and ramification behaviors within local fields, as it connects arithmetic properties directly to their geometric interpretations.
• Evaluate how ramification groups interact with valuation rings and what implications this interaction has for field extensions.
  • Ramification groups interact with valuation rings by providing insight into how primes split or ramify during field extensions. Specifically, within a Galois extension, these groups reveal information about the structure of valuation rings, allowing mathematicians to trace back how valuations change under extension. This relationship has significant implications for understanding algebraic structures, as it connects number theory concepts with broader algebraic frameworks. By studying these interactions, one can derive results about the behavior of prime ideals and their decomposition patterns in extensions.

Related terms

Discrete valuation:

A discrete valuation is a valuation that takes values in the integers and provides a way to measure the 'size' of elements in a field or its extension, determining how many times an element can be divided by a prime element.

Local field: A local field is a field that is complete with respect to a discrete valuation, and it has a finite residue field, making it an essential concept for understanding valuation rings.

Ramification groups: Ramification groups are collections of automorphisms of a Galois extension that provide insights into how primes behave under extensions, particularly in the context of valuations and valuation rings.