🔢Algebraic Number Theory Unit 7 – Prime Decomposition in Number Fields

Foundations of Number Fields

• Number fields extend the concept of rational numbers to include roots of polynomials with rational coefficients
• A number field $K$ is a finite extension of the field of rational numbers $\mathbb{Q}$
  • Every element in $K$ can be expressed as a linear combination of a finite basis over $\mathbb{Q}$
• The degree of a number field $[K:\mathbb{Q}]$ is the dimension of $K$ as a vector space over $\mathbb{Q}$
• Number fields are algebraic extensions of $\mathbb{Q}$, meaning every element is a root of a polynomial with rational coefficients
• The ring of integers $\mathcal{O}_K$ of a number field $K$ consists of elements that are roots of monic polynomials with integer coefficients
  • $\mathcal{O}_K$ is a subring of $K$ and plays a crucial role in studying prime decomposition

Prime Decomposition Basics

• Prime decomposition in number fields generalizes the concept of prime factorization in the integers
• In a number field $K$, prime ideals take the role of prime numbers in the ring of integers $\mathcal{O}_K$
• Every nonzero ideal in $\mathcal{O}_K$ can be uniquely expressed as a product of prime ideals (up to the order of factors)
• The decomposition of a prime number $p \in \mathbb{Z}$ in $\mathcal{O}_K$ is determined by the factorization of the ideal $p\mathcal{O}_K$
  • $p\mathcal{O}_K = \mathfrak{p}_1^{e_1} \cdots \mathfrak{p}_r^{e_r}$, where $\mathfrak{p}_i$ are prime ideals and $e_i$ are positive integers
• The exponents $e_i$ in the prime decomposition are called the ramification indices
• The residue class degree $f_i$ of a prime ideal $\mathfrak{p}_i$ is the degree of the field extension $\mathcal{O}_K/\mathfrak{p}_i$ over $\mathbb{Z}/p\mathbb{Z}$

Unique Factorization in Number Fields

• Unique factorization of elements in a number field $K$ depends on the properties of its ring of integers $\mathcal{O}_K$
• A number field $K$ is called a unique factorization domain (UFD) if every nonzero element in $\mathcal{O}_K$ can be uniquely expressed as a product of irreducible elements (up to the order and units)
• Not all number fields are UFDs; the failure of unique factorization is related to the presence of non-principal ideals in $\mathcal{O}_K$
• The class number $h_K$ of a number field $K$ measures the extent to which unique factorization fails in $\mathcal{O}_K$
  • $h_K = 1$ if and only if $\mathcal{O}_K$ is a UFD
• Examples of number fields that are UFDs include the Gaussian integers $\mathbb{Z}[i]$ and the Eisenstein integers $\mathbb{Z}[\omega]$, where $\omega$ is a primitive third root of unity

Ramification and Splitting

• Ramification occurs when a prime ideal $\mathfrak{p}$ in the ring of integers $\mathcal{O}_K$ appears with an exponent $e > 1$ in the prime decomposition of $p\mathcal{O}_K$
  • The prime $p$ is said to ramify in $K$, and $\mathfrak{p}$ is called a ramified prime
• Splitting refers to the factorization of a prime ideal $p\mathcal{O}_K$ into distinct prime ideals in $\mathcal{O}_K$
  • If $p\mathcal{O}_K = \mathfrak{p}_1 \cdots \mathfrak{p}_r$ with distinct prime ideals $\mathfrak{p}_i$, then $p$ is said to split completely in $K$
• The splitting behavior of primes in a number field is determined by the Dedekind-Kummer theorem
  • It relates the splitting of primes to the factorization of certain polynomials modulo $p$
• Ramification is connected to the discriminant $\Delta_K$ of the number field $K$
  • A prime $p$ ramifies in $K$ if and only if $p$ divides $\Delta_K$
• The study of ramification and splitting is crucial for understanding the arithmetic properties of number fields and their extensions

Ideal Theory in Number Fields

• Ideal theory plays a central role in the study of number fields and their rings of integers
• An ideal $I$ in the ring of integers $\mathcal{O}_K$ is a subset closed under addition and multiplication by elements of $\mathcal{O}_K$
• Principal ideals are generated by a single element $\alpha \in \mathcal{O}_K$, denoted as $(\alpha) = \{\alpha \beta : \beta \in \mathcal{O}_K\}$
• The ideal class group $Cl_K$ of a number field $K$ is the quotient group of fractional ideals modulo principal ideals
  • Its order is the class number $h_K$, which measures the failure of unique factorization in $\mathcal{O}_K$
• The norm of an ideal $I$ is defined as $N(I) = |\mathcal{O}_K/I|$, the size of the quotient ring
  • For a principal ideal $(\alpha)$, the norm is equal to $|N_{K/\mathbb{Q}}(\alpha)|$, where $N_{K/\mathbb{Q}}$ is the field norm
• Ideal theory allows for a generalization of the unique factorization theorem to all number fields
  • Every nonzero ideal in $\mathcal{O}_K$ can be uniquely expressed as a product of prime ideals

Dedekind Domains and Prime Ideals

• The ring of integers $\mathcal{O}_K$ of a number field $K$ is a Dedekind domain
  • Dedekind domains are integral domains where every nonzero ideal can be uniquely factored into a product of prime ideals
• In a Dedekind domain, prime ideals are maximal ideals, and every nonzero prime ideal is maximal
• The localization of a Dedekind domain at a prime ideal is a discrete valuation ring
  • This property allows for the definition of valuations and completions of number fields
• The prime ideals in $\mathcal{O}_K$ lie above prime numbers in $\mathbb{Z}$
  • The lying above relation is characterized by the prime decomposition of $p\mathcal{O}_K$ for prime numbers $p$
• The Dedekind-Kummer theorem provides a criterion for the splitting behavior of prime ideals in terms of the factorization of polynomials modulo $p$
• The study of prime ideals in Dedekind domains is fundamental to understanding the arithmetic and geometric properties of number fields

Applications and Examples

• Fermat's Last Theorem: The proof by Andrew Wiles relies heavily on the theory of elliptic curves and modular forms over number fields
• Cryptography: Number fields and their prime ideals are used in various cryptographic schemes, such as the Buchmann-Williams key exchange and the Gentry-Szydlo algorithm
• Class field theory: It describes the abelian extensions of a number field in terms of its ideal class group and idele class group
  • The Hilbert class field of $K$ is the maximal unramified abelian extension of $K$
• Diophantine equations: Many Diophantine equations, such as the Pell equation and the Thue equation, can be studied using the arithmetic of number fields
• Algebraic number theory: Prime decomposition is a fundamental tool in the study of zeta functions, L-functions, and arithmetic geometry
• Examples of number fields:
  • Quadratic fields: $\mathbb{Q}(\sqrt{d})$, where $d$ is a squarefree integer (e.g., $\mathbb{Q}(\sqrt{2})$, $\mathbb{Q}(\sqrt{-5})$)
  • Cyclotomic fields: $\mathbb{Q}(\zeta_n)$, where $\zeta_n$ is a primitive $n$-th root of unity (e.g., $\mathbb{Q}(i)$, $\mathbb{Q}(\omega)$)

Advanced Topics and Open Problems

• Iwasawa theory: It studies the behavior of prime ideals and class groups in infinite towers of number fields, such as $\mathbb{Z}_p$-extensions
• Langlands program: It seeks to unify various areas of mathematics, including number theory, representation theory, and harmonic analysis, through the study of automorphic forms and Galois representations
• Stark conjectures: They relate the values of L-functions to the arithmetic of number fields, providing a generalization of the analytic class number formula
• ABC conjecture: It states an inequality involving the prime factors of three relatively prime integers $a$, $b$, and $c$ satisfying $a + b = c$
  • The conjecture has significant implications for the study of Diophantine equations and the distribution of prime numbers
• Fermat-Catalan conjecture: It generalizes Fermat's Last Theorem to other powers and seeks to classify all solutions to the equation $a^m + b^n = c^k$ in positive integers with $\frac{1}{m} + \frac{1}{n} + \frac{1}{k} < 1$
• Nonabelian class field theory: It aims to extend the results of class field theory to nonabelian extensions of number fields
  • This area is still largely conjectural and remains an active area of research in modern number theory