🔢Algebraic Number Theory Unit 12 – Valuations and Ramification in Number Theory

Key Concepts and Definitions

• Valuation a function $v: K \to \mathbb{R} \cup \{\infty\}$ satisfying certain properties (non-negativity, multiplicativity, and triangle inequality)
• Absolute value a special case of valuation, often denoted as $|\cdot|$
  • Examples include the usual absolute value on $\mathbb{Q}$ and $p$-adic absolute value on $\mathbb{Q}$
• Discrete valuation a valuation with discrete image, often normalized so that the image is $\mathbb{Z} \cup \{\infty\}$
• Completion of a field $K$ with respect to a valuation $v$, denoted as $K_v$, a field containing $K$ where $v$ extends uniquely
• Ring of integers in a number field $K$, denoted as $\mathcal{O}_K$, consists of elements $x \in K$ such that $v(x) \geq 0$ for all discrete valuations $v$ on $K$
• Prime ideal in $\mathcal{O}_K$ a non-zero proper ideal $\mathfrak{p}$ such that if $ab \in \mathfrak{p}$, then either $a \in \mathfrak{p}$ or $b \in \mathfrak{p}$
• Ramification the phenomenon where a prime ideal in a base field (e.g., $\mathbb{Q}$) splits into multiple prime ideals in an extension field (e.g., a number field $K$)

Valuation Theory Foundations

• Valuations generalize the notion of absolute value and provide a way to measure the "size" of elements in a field
• Properties of valuations include non-negativity ($v(x) \geq 0$), multiplicativity ($v(xy) = v(x) + v(y)$), and triangle inequality ($v(x+y) \geq \min\{v(x), v(y)\}$)
  • These properties allow valuations to behave like a "distance" on a field
• Valuations can be used to define a topology on a field, making it a topological field
  • Open sets are defined as sets of the form $\{x \in K : v(x-a) > \varepsilon\}$ for some $a \in K$ and $\varepsilon > 0$
• Completions of fields with respect to valuations are important in number theory and algebraic geometry
  • Examples include the real numbers $\mathbb{R}$ (completion of $\mathbb{Q}$ with respect to the usual absolute value) and the $p$-adic numbers $\mathbb{Q}_p$ (completion of $\mathbb{Q}$ with respect to the $p$-adic valuation)
• Valuations can be extended from a base field to an extension field, but the extension may not be unique
• The set of all valuations on a field forms a partially ordered set under the relation $v_1 \leq v_2$ if $v_1(x) \leq v_2(x)$ for all $x$

Types of Valuations in Number Fields

• Discrete valuations are valuations with discrete image, often normalized to have image $\mathbb{Z} \cup \{\infty\}$
  • In number fields, discrete valuations correspond to prime ideals in the ring of integers
• Archimedean valuations are valuations that satisfy the Archimedean property: for any $x, y > 0$, there exists an integer $n$ such that $v(nx) > v(y)$
  • Examples include the usual absolute value on $\mathbb{Q}$ and its extensions to number fields
• Non-Archimedean valuations are valuations that do not satisfy the Archimedean property
  • Examples include the $p$-adic valuations on $\mathbb{Q}$ and their extensions to number fields
• Valuations can be classified as either finite (non-zero on a finite number of elements) or infinite (non-zero on an infinite number of elements)
• The product formula relates the valuations on a number field: for any non-zero $x \in K$, we have $\prod_v |x|_v = 1$, where the product is taken over all normalized valuations on $K$
  • This formula has important applications in number theory, such as proving the finiteness of the class number

Ramification in Number Fields

• Ramification occurs when a prime ideal in a base field (e.g., $\mathbb{Q}$) splits into multiple prime ideals in an extension field (e.g., a number field $K$)
  • The prime ideals in the extension field are called the "primes above" the original prime ideal
• The ramification index of a prime ideal $\mathfrak{p}$ in $\mathcal{O}_K$ over a prime $p$ in $\mathbb{Z}$ is the exponent $e$ such that $p\mathcal{O}_K = \mathfrak{p}^e \cdot \mathfrak{a}$, where $\mathfrak{a}$ is an ideal coprime to $\mathfrak{p}$
  • Intuitively, the ramification index measures how much a prime "ramifies" in the extension
• A prime is said to be unramified if its ramification index is 1, and ramified otherwise
• The ramification of primes is related to the discriminant of a number field, which measures the "size" of the ring of integers
  • A number field is said to be unramified at a prime $p$ if $p$ does not divide the discriminant
• Ramification is an important concept in the study of extensions of local fields (completions of number fields with respect to a prime ideal)
  • The ramification index and inertia degree of a local field extension provide information about the structure of the extension

Local and Global Fields

• Local fields are complete fields with respect to a discrete valuation
  • Examples include the $p$-adic numbers $\mathbb{Q}_p$ and finite extensions of $\mathbb{Q}_p$
• Global fields are either number fields (finite extensions of $\mathbb{Q}$) or function fields (finite extensions of $\mathbb{F}_p(t)$)
• The study of local fields is closely related to the study of valuations and completions
  • Many properties of number fields can be understood by studying their completions with respect to different valuations
• The local-global principle, also known as the Hasse principle, relates the behavior of equations over local fields to their behavior over global fields
  • For example, the Hasse-Minkowski theorem states that a quadratic form over a global field has a non-zero solution if and only if it has a non-zero solution over all completions of the field
• Adeles and ideles are important constructions in the study of global fields, providing a way to simultaneously consider all completions of a field
  • The adeles of a global field are the restricted product of its completions with respect to all valuations, while the ideles are the multiplicative group of the adeles

Applications to Algebraic Number Theory

• Valuations and ramification play a crucial role in the study of algebraic number theory, which is concerned with the properties of number fields and their rings of integers
• The decomposition of primes in extensions of number fields is closely related to the ramification of valuations
  • The Dedekind-Kummer theorem describes the decomposition of primes in terms of the factorization of certain polynomials
• The ideal class group of a number field, which measures the failure of unique factorization in the ring of integers, is related to the ramification of primes
  • The Hilbert class field of a number field is the maximal unramified abelian extension of the field
• The study of valuations and completions is important in the development of local class field theory, which describes abelian extensions of local fields
  • The Artin reciprocity law relates the Galois group of an abelian extension to certain subgroups of the idele class group
• Valuations and ramification also play a role in the study of elliptic curves and other geometric objects defined over number fields
  • The reduction of an elliptic curve modulo a prime ideal is related to the ramification of the prime in the field of definition

Problem-Solving Techniques

• When working with valuations, it's important to keep track of their properties, such as non-negativity, multiplicativity, and the triangle inequality
  • These properties can be used to simplify expressions and prove inequalities
• When studying ramification, it's helpful to consider the factorization of prime ideals in the base field and the extension field
  • The ramification index and inertia degree provide information about how the primes split in the extension
• Local-global principles, such as the Hasse principle, can be used to reduce problems over global fields to problems over local fields
  • This can often simplify the problem, as local fields have a simpler structure than global fields
• The use of adeles and ideles can help in the study of global fields by providing a way to consider all completions simultaneously
  • Many problems in algebraic number theory can be formulated in terms of adeles and ideles
• When working with extensions of number fields, it's important to consider the Galois group of the extension and its subgroups
  • The Galois correspondence relates subgroups of the Galois group to intermediate fields of the extension
• Computational tools, such as computer algebra systems like Magma or SageMath, can be helpful in exploring examples and testing conjectures
  • These tools can perform complex calculations with number fields, ideals, and valuations

Further Exploration and Open Questions

• The study of higher-dimensional local fields, which are complete with respect to a higher-dimensional valuation, is an active area of research
  • These fields have applications in algebraic geometry and the study of higher-dimensional schemes
• The theory of perfectoid spaces, developed by Peter Scholze, provides a way to relate certain geometric objects in characteristic p to objects in characteristic 0
  • Perfectoid spaces have led to significant advances in the study of p-adic geometry and the Langlands program
• The study of p-adic Hodge theory, which relates p-adic Galois representations to objects in p-adic analysis and geometry, is an important area of research
  • The theory has applications in the study of elliptic curves, modular forms, and other geometric objects
• The Langlands program, which relates Galois representations to automorphic forms, is a major open problem in number theory
  • The study of valuations and ramification plays a role in the formulation and study of the Langlands conjectures
• The abc conjecture, which relates the prime factorization of three integers $a$, $b$, and $c$ satisfying $a + b = c$, has important consequences in number theory
  • The conjecture is related to the study of valuations and ramification in number fields
• The study of non-abelian extensions of number fields, and their relation to valuations and ramification, is an active area of research
  • Non-abelian class field theory, which aims to describe all finite extensions of number fields, is a major open problem in algebraic number theory