12.1 Discrete valuations and valuation rings
3 min read•july 30, 2024
Discrete valuations are powerful tools in algebraic number theory, measuring the "size" of elements in a field. They map field elements to integers, satisfying specific axioms and inducing a unique topology. These valuations are closely tied to prime ideals in Dedekind domains.
Valuation rings, containing elements with non-negative valuation, play a crucial role in understanding field structures. They're principal ideal domains with a single , forming the backbone of local field theory. This connection to bridges global and local perspectives in number theory.
Discrete valuations and properties
Definition and core characteristics
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- Discrete valuation functions map field K to integers Z union {infinity}
- Satisfy axioms and
- Image forms discrete subgroup of real numbers (typically isomorphic to integers)
- isomorphic to Z
- Elements in field uniquely expressed as (u unit, n integer)
- Induce topology on field (basis of neighborhoods of 0 formed by sets )
Key properties and distinctions
- Possess unique maximal ideal generated by uniformizer π where
- Exhibit ultrametric inequality (distinguishes from other valuation types)
- Correspond to nonzero prime ideals in Dedekind domains (one-to-one relationship)
- Satisfy strong triangle inequality in induced absolute value
Examples and applications
- on rational numbers (measures divisibility by prime p)
- Valuation on polynomial ring K[x] (measures degree of polynomials)
- Discrete valuation on function field of algebraic curve (corresponds to points on curve)
Valuation rings and ideals
Structure of valuation rings
- R of field K contains elements x where
- Constitute principal ideal domains (PIDs) with unique non-zero prime ideal
- Function as local rings (possess single maximal ideal)
- Fraction field equals field K on which valuation defined
- Completion with respect to maximal ideal yields ring
Ideal properties
- Maximal ideal M comprises elements x where
- All proper ideals take form (n non-negative integer, π uniformizer)
- Units of ring R precisely elements x where
- Quotient of ring by maximal ideal forms residue field
Examples and applications
- Ring of integers in a number field (localized at a prime ideal)
- Formal power series ring k[[t]] over field k
- Local ring of regular point on algebraic variety
Structure of valuation rings
Algebraic properties
- Integrally closed in their quotient fields
- Quotient field represents smallest field containing the ring
- Non-zero elements in quotient field uniquely expressed as (u unit in valuation ring, n integer)
- Completion of quotient field with respect to valuation topology forms complete discrete valuation field
Analytical tools
- Hensel's Lemma enables lifting polynomial equation solutions from residue field to valuation ring
- Utilized for finding roots of polynomials over
- Allows construction of unramified extensions of local fields
Examples and applications
- Z(p) (integers localized at prime p) as valuation ring of Q
- k[[t]] (formal power series) as valuation ring of k((t)) (formal Laurent series)
- Valuation ring of completion of algebraic number field at prime ideal
Valuations vs local fields
Characteristics of local fields
- Complete fields with respect to discrete valuation
- Ring of integers forms complete discrete valuation ring
- Classified into characteristic 0 (finite extensions of p-adic numbers) and positive characteristic (formal Laurent series over finite fields)
- Always possess finite residue field
Ramification and extensions
- Ramification theory examines decomposition of prime ideals in local field extensions
- Local class field theory describes abelian extensions using multiplicative group
- Unramified extensions correspond to extensions of residue field
Examples and applications
- Qp (p-adic numbers) as local field of characteristic 0
- Fp((t)) (formal Laurent series over finite field) as local field of positive characteristic
- Completion of number field at prime ideal (used in studying local properties of global fields)
Key Terms to Review (16)
Complete discrete valuation: A complete discrete valuation is a type of valuation that assigns values to elements of a field in such a way that it satisfies the properties of a discrete valuation while also being complete with respect to a topology induced by the valuation. This means that every Cauchy sequence converges within the field, highlighting the relationship between the algebraic structure and topological completeness, which is essential in understanding valuation rings and their properties.
Composite valuation: Composite valuation refers to a valuation that arises from combining multiple discrete valuations on a given field or ring. This concept is important because it helps understand how different valuations can interact and influence the overall structure of a valuation ring, especially in algebraic number theory. By examining composite valuations, one can gain insights into how various algebraic properties are affected by the interplay between different discrete valuations.
Extension of a valuation: 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.
K[x]_{(x)}: The notation k[x]_{(x)} represents the localization of the polynomial ring k[x] at the prime ideal generated by the polynomial x. This construction allows for a focus on polynomials that have roots in the vicinity of x = 0, providing a way to study the local properties of these polynomials and their behavior in a neighborhood around that point.
Krull's Theorem: Krull's Theorem states that for any Noetherian ring, the set of prime ideals can be arranged in a hierarchy known as the prime spectrum, where each prime ideal corresponds to a unique valuation. This theorem is essential for understanding how discrete valuations and valuation rings interact within the framework of algebraic number theory, especially in characterizing the structure of these rings and their ideals.
Local fields: Local fields are a special class of fields that are complete with respect to a discrete valuation and have finite residue fields. These fields are crucial in number theory, particularly when studying properties of p-adic numbers and how they behave under various arithmetic operations. They also serve as foundational elements when looking at discrete valuations and their corresponding valuation rings, as well as in the context of understanding the Artin reciprocity law, which links local and global fields through Galois theory.
Maximal Ideal: A maximal ideal is an ideal in a ring that is proper and maximal with respect to inclusion, meaning it cannot be contained in any larger proper ideal. This concept connects to important structures in algebraic systems, helping in the classification of rings and understanding algebraic integers and numbers, as well as paving the way to comprehend prime ideals and their unique properties within a ring.
Ostrowski's Theorem: Ostrowski's Theorem states that every non-archimedean absolute value on a number field is equivalent to either a discrete valuation or the p-adic valuation for some prime p. This theorem connects the study of valuations, completions of number fields, and the structure of local fields. It plays a crucial role in understanding how number fields can be completed and analyzed through their valuations.
P-adic numbers: p-adic numbers are a system of numbers used in number theory that extend the ordinary arithmetic of rational numbers by introducing a new way of measuring distances, based on a prime number p. This approach allows mathematicians to consider convergence and limits in a different framework, leading to new insights into divisibility and number properties. They provide an essential tool for studying various algebraic structures and can be utilized to solve equations with unique properties in modular arithmetic.
P-adic valuation: The p-adic valuation is a function that assigns to each non-zero rational number a non-negative integer, reflecting how many times that number can be divided by a prime number p before it becomes a unit. This concept is essential in understanding the structure of p-adic numbers and their fields, as it provides a way to measure the 'size' of numbers in a different sense than the usual absolute value. It links deeply with discrete valuations and valuation rings, playing a crucial role in algebraic number theory by providing a means to study the properties of integers and rational numbers in relation to prime factors.
Trivial valuation: A trivial valuation is a specific type of valuation that assigns the value of zero to every non-zero element of a field, while the value of zero itself is typically assigned infinity. This means that all non-zero elements are considered equivalent in terms of their valuation, emphasizing that they do not contribute to any hierarchy or ordering in the valuation. In the context of discrete valuations and valuation rings, trivial valuations serve as a baseline comparison, highlighting the characteristics of more nuanced valuations.
Valuation Domains: Valuation domains are integral domains that are characterized by the existence of a valuation, which assigns a value to each element in a way that captures the 'size' or 'multiplicative structure' of the elements. This concept connects to discrete valuations and valuation rings as it helps in understanding the structure of these rings, allowing for a classification of their ideals and facilitating the study of local properties in algebraic number theory.
Valuation map: 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.
Valuation ring: 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.
Value group: The value group is an essential concept in the context of discrete valuations and valuation rings, representing the set of values assigned to elements in a field with respect to a chosen valuation. This group typically takes the form of an ordered abelian group, where the values correspond to the 'size' or 'degree of divisibility' of elements. It plays a crucial role in understanding how different elements relate to one another in terms of their valuation and provides insights into the structure of the associated valuation ring.
Z_p: The symbol $$\mathbb{Z}_p$$ denotes the ring of p-adic integers, which is a complete metric space with respect to the p-adic valuation. This ring is crucial in number theory as it allows for an alternative way of looking at integers and provides a framework for discussing convergence and limits in a unique way that is different from the usual real number system.