🔢Algebraic Number Theory Unit 2 – Number Fields and Integer Rings

Key Concepts and Definitions

• Number field $K$ is a finite extension of the field of rational numbers $\mathbb{Q}$
• Degree of a number field $[K:\mathbb{Q}]$ is the dimension of $K$ as a vector space over $\mathbb{Q}$
• Algebraic number is a complex number that is a root of a non-zero polynomial with rational coefficients
• Algebraic integer is an algebraic number that is a root of a monic polynomial with integer coefficients
• Ring of integers $\mathcal{O}_K$ of a number field $K$ consists of all algebraic integers in $K$
  • Example: In the number field $\mathbb{Q}(\sqrt{5})$, the ring of integers is $\mathbb{Z}[\frac{1+\sqrt{5}}{2}]$
• Ideal in a ring $R$ is a subset $I \subseteq R$ closed under addition and absorption by ring elements
• Principal ideal is an ideal generated by a single element $a \in R$, denoted as $(a) = {ra : r \in R}$
• Unique factorization domain (UFD) is an integral domain where every non-zero element can be uniquely factored into irreducible elements up to ordering and unit factors

Number Fields: Foundation and Properties

• Number field $K$ is a field extension of $\mathbb{Q}$ of finite degree $n = [K:\mathbb{Q}]$
• Every number field $K$ can be obtained by adjoining a single algebraic number $\alpha$ to $\mathbb{Q}$, i.e., $K = \mathbb{Q}(\alpha)$
• Minimal polynomial of an algebraic number $\alpha$ is the monic irreducible polynomial $m_\alpha(x) \in \mathbb{Q}[x]$ of lowest degree such that $m_\alpha(\alpha) = 0$
  • Degree of $\alpha$ is the degree of its minimal polynomial
• Primitive element theorem states that every number field $K$ has a primitive element $\alpha$ such that $K = \mathbb{Q}(\alpha)$
• Embeddings of a number field $K$ into $\mathbb{C}$ are field homomorphisms $\sigma: K \rightarrow \mathbb{C}$ fixing $\mathbb{Q}$
  • Number of embeddings equals the degree of the number field
• Trace and norm of an element $\alpha \in K$ are defined as:
  • $Tr_{K/\mathbb{Q}}(\alpha) = \sum_{\sigma} \sigma(\alpha)$
  • $N_{K/\mathbb{Q}}(\alpha) = \prod_{\sigma} \sigma(\alpha)$ where $\sigma$ runs through all embeddings of $K$ into $\mathbb{C}$

Integer Rings in Number Fields

• Ring of integers $\mathcal{O}_K$ of a number field $K$ is the set of all algebraic integers in $K$
• $\mathcal{O}_K$ is a subring of $K$ and a free $\mathbb{Z}$-module of rank equal to the degree of $K$
• Integral basis of $\mathcal{O}_K$ is a basis of $\mathcal{O}_K$ as a $\mathbb{Z}$-module
  • Example: In $\mathbb{Q}(\sqrt{2})$, an integral basis is ${1, \sqrt{2}}$
• Discriminant of a number field $K$ is defined as $\Delta_K = \det(Tr_{K/\mathbb{Q}}(\alpha_i\alpha_j))_{1 \leq i,j \leq n}$, where ${\alpha_1, \ldots, \alpha_n}$ is an integral basis of $\mathcal{O}_K$
• Dedekind domain is an integral domain where every ideal can be uniquely factored into prime ideals
  • Ring of integers $\mathcal{O}_K$ is a Dedekind domain
• Fractional ideal of $\mathcal{O}_K$ is a non-zero $\mathcal{O}_K$-submodule of $K$
  • Every fractional ideal is invertible, i.e., for every fractional ideal $I$, there exists a fractional ideal $J$ such that $IJ = \mathcal{O}_K$

Algebraic Integers and Their Properties

• Algebraic integer is an algebraic number that satisfies a monic polynomial with integer coefficients
• Sum, difference, and product of algebraic integers are also algebraic integers
• Every algebraic integer belongs to the ring of integers $\mathcal{O}_K$ of some number field $K$
• Norm and trace of an algebraic integer are integers
  • For $\alpha \in \mathcal{O}K$, $Tr{K/\mathbb{Q}}(\alpha) \in \mathbb{Z}$ and $N_{K/\mathbb{Q}}(\alpha) \in \mathbb{Z}$
• Algebraic integers in a number field $K$ form a ring, which is the ring of integers $\mathcal{O}_K$
• Units in $\mathcal{O}_K$ are algebraic integers with norm $\pm 1$
  • Example: In $\mathbb{Z}[\sqrt{2}]$, the units are $\pm 1$
• Dirichlet's unit theorem states that the group of units in $\mathcal{O}_K$ is finitely generated

Ideals in Integer Rings

• Ideal in the ring of integers $\mathcal{O}_K$ is an additive subgroup closed under multiplication by elements of $\mathcal{O}_K$
• Principal ideal in $\mathcal{O}_K$ is an ideal generated by a single element $\alpha \in \mathcal{O}_K$, denoted as $(\alpha) = {\alpha\beta : \beta \in \mathcal{O}_K}$
• Prime ideal $\mathfrak{p}$ in $\mathcal{O}_K$ is a proper ideal such that for any $\alpha, \beta \in \mathcal{O}_K$, if $\alpha\beta \in \mathfrak{p}$, then either $\alpha \in \mathfrak{p}$ or $\beta \in \mathfrak{p}$
• Maximal ideal in $\mathcal{O}_K$ is a proper ideal that is not contained in any other proper ideal
  • Every maximal ideal is prime, but not every prime ideal is maximal
• Norm of an ideal $I$ in $\mathcal{O}_K$ is defined as $N(I) = |\mathcal{O}_K/I|$, the cardinality of the quotient ring
  • For a principal ideal $(\alpha)$, $N((\alpha)) = |N_{K/\mathbb{Q}}(\alpha)|$
• Sum and product of ideals in $\mathcal{O}_K$ are also ideals in $\mathcal{O}_K$

Factorization and Unique Factorization Domains

• Unique factorization domain (UFD) is an integral domain where every non-zero element can be uniquely factored into irreducible elements up to ordering and unit factors
• Ring of integers $\mathcal{O}_K$ is a UFD if and only if every ideal in $\mathcal{O}_K$ is principal
  • Example: $\mathbb{Z}[\sqrt{-5}]$ is not a UFD, as the ideal $(2, 1+\sqrt{-5})$ is not principal
• In a Dedekind domain, every ideal can be uniquely factored into prime ideals
  • Ring of integers $\mathcal{O}_K$ is a Dedekind domain, so every ideal in $\mathcal{O}_K$ has a unique factorization into prime ideals
• Class number of a number field $K$ is the cardinality of the ideal class group, which measures the failure of unique factorization in $\mathcal{O}_K$
  • $K$ has class number 1 if and only if $\mathcal{O}_K$ is a UFD
• Euclidean domain is an integral domain with a Euclidean function, which implies it is a UFD
  • Example: $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain with the Euclidean function $N(a+b\sqrt{2}) = |a^2 - 2b^2|$

Applications and Examples

• Number fields and their rings of integers have applications in cryptography, such as in the construction of lattice-based cryptosystems
  • Example: The ring of integers of a cyclotomic field can be used to construct a lattice for the NTRU cryptosystem
• Algebraic number theory is used in the study of Diophantine equations, which are polynomial equations with integer coefficients and integer solutions
  • Example: Fermat's Last Theorem states that the equation $x^n + y^n = z^n$ has no non-zero integer solutions for $n > 2$
• Class number formula relates the class number of a number field to its discriminant and the values of its Dedekind zeta function
  • Example: For a quadratic field $K = \mathbb{Q}(\sqrt{d})$, the class number $h_K$ satisfies $h_K = \frac{\sqrt{|\Delta_K|}}{2\pi}L(1, \chi_d)$, where $L(s, \chi_d)$ is the Dirichlet L-function associated with the quadratic character $\chi_d$
• Primes in $\mathbb{Z}$ may factor into prime ideals in the ring of integers of a number field
  • Example: In $\mathbb{Z}[\sqrt{-5}]$, the prime 2 factors as $(2) = (2, 1+\sqrt{-5})(2, 1-\sqrt{-5})$
• Units in the ring of integers form a finitely generated abelian group, as described by Dirichlet's unit theorem
  • Example: In $\mathbb{Q}(\sqrt{2})$, the units are $\pm(1+\sqrt{2})^n$ for $n \in \mathbb{Z}$

Advanced Topics and Extensions

• Dedekind zeta function of a number field $K$ is a generalization of the Riemann zeta function, defined as $\zeta_K(s) = \sum_{\mathfrak{a}} \frac{1}{N(\mathfrak{a})^s}$, where $\mathfrak{a}$ runs through all non-zero ideals of $\mathcal{O}_K$
  • Analytic class number formula expresses the residue of $\zeta_K(s)$ at $s=1$ in terms of the class number, regulator, and other invariants of $K$
• Artin reciprocity law is a central result in class field theory, relating the abelian extensions of a number field to its idele class group
  • Ideles of a number field $K$ are elements of the restricted product of the completions of $K$ with respect to its non-archimedean absolute values
• Langlands program is a vast network of conjectures connecting representation theory, automorphic forms, and arithmetic geometry
  • Langlands reciprocity conjecture relates Galois representations to automorphic representations of reductive groups over number fields
• Elliptic curves over number fields have a rich arithmetic structure and are connected to various problems in number theory
  • Mordell-Weil theorem states that the group of $K$-rational points on an elliptic curve over a number field $K$ is finitely generated
• Iwasawa theory studies the growth of arithmetic objects (such as class groups) in towers of number fields
  • Main conjecture of Iwasawa theory relates the characteristic ideal of the Iwasawa module to the p-adic L-function
• Stark conjectures provide a conjectural description of the leading term of the Taylor expansion of Artin L-functions at $s=0$ in terms of units in number fields