🔢Algebraic Number Theory Unit 4 – Norms, Traces, and Discriminants

Key Concepts and Definitions

• Algebraic number theory studies algebraic numbers, which are roots of polynomials with integer coefficients
• Algebraic integers are complex numbers that are roots of monic polynomials with integer coefficients
• Number fields are finite extensions of the field of rational numbers $\mathbb{Q}$
  • Examples include $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(i)$
• Ring of integers $\mathcal{O}_K$ of a number field $K$ consists of all algebraic integers in $K$
• Norm $N_{K/\mathbb{Q}}(\alpha)$ of an element $\alpha$ in a number field $K$ is the product of its conjugates
• Trace $Tr_{K/\mathbb{Q}}(\alpha)$ of an element $\alpha$ in a number field $K$ is the sum of its conjugates
• Discriminant $\Delta_K$ of a number field $K$ measures the complexity of its ring of integers $\mathcal{O}_K$

Historical Context and Development

• Early work in algebraic number theory dates back to Fermat, Euler, and Gauss in the 17th-19th centuries
• Kummer introduced ideal numbers in the 1840s to study higher reciprocity laws and Fermat's Last Theorem
• Dedekind and Kronecker independently developed the concept of ideals in the 1870s
  • Ideals generalize the notion of divisibility in rings
• Hilbert's Zahlbericht (1897) provided a comprehensive treatment of algebraic number theory
• Class field theory, developed in the early 20th century, describes abelian extensions of number fields
• Langlands program, proposed in the 1960s, aims to unify various areas of mathematics through automorphic forms and Galois representations

Norms in Algebraic Number Theory

• Norm of an element $\alpha$ in a number field $K$ is defined as $N_{K/\mathbb{Q}}(\alpha) = \prod_{\sigma} \sigma(\alpha)$, where $\sigma$ runs over all embeddings of $K$ into $\mathbb{C}$
• Norm is multiplicative: $N_{K/\mathbb{Q}}(\alpha\beta) = N_{K/\mathbb{Q}}(\alpha)N_{K/\mathbb{Q}}(\beta)$
• Norm of an algebraic integer is always an integer
• Norm of a unit (invertible element) in the ring of integers $\mathcal{O}_K$ is $\pm 1$
• Norm can be used to define the ideal norm $N(\mathfrak{a}) = |\mathcal{O}_K/\mathfrak{a}|$ for an ideal $\mathfrak{a}$ in $\mathcal{O}_K$
• Ideal norm is multiplicative: $N(\mathfrak{ab}) = N(\mathfrak{a})N(\mathfrak{b})$
• Norm plays a crucial role in the factorization of ideals and the study of class groups

Traces and Their Properties

• Trace of an element $\alpha$ in a number field $K$ is defined as $Tr_{K/\mathbb{Q}}(\alpha) = \sum_{\sigma} \sigma(\alpha)$, where $\sigma$ runs over all embeddings of $K$ into $\mathbb{C}$
• Trace is additive: $Tr_{K/\mathbb{Q}}(\alpha + \beta) = Tr_{K/\mathbb{Q}}(\alpha) + Tr_{K/\mathbb{Q}}(\beta)$
• Trace is $\mathbb{Q}$-linear: $Tr_{K/\mathbb{Q}}(c\alpha) = cTr_{K/\mathbb{Q}}(\alpha)$ for $c \in \mathbb{Q}$
• Trace of an algebraic integer is always an integer
• Transitivity of trace: if $L/K/\mathbb{Q}$ is a tower of field extensions, then $Tr_{L/\mathbb{Q}}(\alpha) = Tr_{K/\mathbb{Q}}(Tr_{L/K}(\alpha))$
• Trace and norm are related by the Cayley-Hamilton theorem: if $\alpha \in K$ and $f(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_0$ is its minimal polynomial over $\mathbb{Q}$, then $a_{n-1} = -Tr_{K/\mathbb{Q}}(\alpha)$ and $a_0 = (-1)^nN_{K/\mathbb{Q}}(\alpha)$

Discriminants: Fundamentals and Applications

• Discriminant $\Delta_K$ of a number field $K$ is defined as the determinant of the trace form $Tr_{K/\mathbb{Q}}(\alpha_i\alpha_j)$, where $\{\alpha_1, \ldots, \alpha_n\}$ is an integral basis of $\mathcal{O}_K$
• Discriminant measures the complexity of the ring of integers $\mathcal{O}_K$
  • Larger absolute values of the discriminant indicate a more complex ring of integers
• Discriminant is related to ramification in number fields
  • Prime numbers dividing the discriminant are precisely the primes that ramify in $K$
• Minkowski's bound states that there exists an ideal $\mathfrak{a}$ in $\mathcal{O}_K$ with norm $N(\mathfrak{a}) \leq \left(\frac{4}{\pi}\right)^{r_2}\frac{n!}{n^n}\sqrt{|\Delta_K|}$, where $n = [K:\mathbb{Q}]$ and $r_2$ is the number of complex embeddings of $K$
• Hermite's constant $\gamma_n$ provides a sharper bound for the norm of an ideal in a number field
• Discriminants play a key role in the study of integral bases, class numbers, and unit groups of number fields

Computational Techniques and Examples

• Computing norms and traces can be done by finding the roots of the minimal polynomial and applying the definitions
  • Example: for $\alpha = \sqrt{2} + i$ in $\mathbb{Q}(\sqrt{2}, i)$, $N(\alpha) = (\sqrt{2} + i)(\sqrt{2} - i)(-\sqrt{2} + i)(-\sqrt{2} - i) = 9$ and $Tr(\alpha) = (\sqrt{2} + i) + (\sqrt{2} - i) + (-\sqrt{2} + i) + (-\sqrt{2} - i) = 0$
• Discriminants can be computed using resultants or determinants of trace matrices
  • Example: for $K = \mathbb{Q}(\sqrt{5})$, an integral basis is $\{1, \frac{1 + \sqrt{5}}{2}\}$, and the discriminant is $\Delta_K = \begin{vmatrix} Tr(1) & Tr(\frac{1 + \sqrt{5}}{2}) \\ Tr(\frac{1 + \sqrt{5}}{2}) & Tr((\frac{1 + \sqrt{5}}{2})^2) \end{vmatrix} = \begin{vmatrix} 2 & 1 \\ 1 & 3 \end{vmatrix} = 5$
• Pari/GP, SageMath, and Magma are popular computational algebra systems for working with algebraic number theory
• Efficient algorithms exist for computing class groups, unit groups, and solving Diophantine equations in number fields

Connections to Other Areas of Mathematics

• Algebraic number theory has close ties to algebraic geometry through the study of schemes and arithmetic geometry
  • Spec$(\mathcal{O}_K)$ is an affine scheme whose geometry encodes arithmetic properties of $K$
• Elliptic curves and abelian varieties over number fields are central objects in arithmetic geometry
  • Mordell-Weil theorem describes the structure of rational points on elliptic curves
• Zeta functions of number fields, such as the Dedekind zeta function, connect algebraic number theory to analytic number theory and complex analysis
• Galois representations and automorphic forms link algebraic number theory to representation theory and harmonic analysis
• Iwasawa theory studies the behavior of class groups and unit groups in towers of number fields
• Stark conjectures relate special values of L-functions to arithmetic data in number fields

Advanced Topics and Current Research

• Langlands program aims to establish reciprocity laws between Galois representations and automorphic representations
  • Taniyama-Shimura conjecture (now a theorem) states that every elliptic curve over $\mathbb{Q}$ is modular
• Birch and Swinnerton-Dyer conjecture relates the rank of an elliptic curve to the order of vanishing of its L-function at $s = 1$
• Iwasawa main conjecture describes the structure of Selmer groups and p-adic L-functions in towers of number fields
• Equivariant Tamagawa number conjecture (ETNC) generalizes the Birch and Swinnerton-Dyer conjecture to motives
• Noncommutative Iwasawa theory studies the structure of Selmer groups over noncommutative p-adic Lie extensions
• Euler systems, such as Heegner points and Stark-Heegner points, are powerful tools for studying the arithmetic of Galois representations
• Anabelian geometry, initiated by Grothendieck, aims to recover arithmetic information from the étale fundamental group of a scheme