Title: Algebraic Number Theory

Algebraic Number Theory is a branch of mathematics that sits at the intersection of algebra and number theory. It extends the study of the integers to more general number systems that are solutions to polynomial equations with integer coefficients. By doing so, it introduces algebraic structures that provide profound insights into the properties of numbers.

Overview

At its core, algebraic number theory deals with algebraic numbers, which are roots of non-zero polynomial equations with integer coefficients. More formally, an algebraic number is a complex number \( \alpha \) such that there exists a polynomial \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \) with integer coefficients \( a_i \) (where \( a_n \neq 0 \)), for which \( f(\alpha) = 0 \).

Algebraic Integers

A subset of algebraic numbers, known as algebraic integers, are roots of monic polynomials (polynomials where the leading coefficient \( a_n \) is \( 1 \)) with integer coefficients. This is analogous to the familiar integers in the set of rational numbers. For instance, \( \alpha \) is an algebraic integer if it satisfies an equation of the form:
\[ \alpha^n + b_{n-1} \alpha^{n-1} + \cdots + b_0 = 0 \]
where \( b_i \) are integers.

Number Fields

A crucial concept in algebraic number theory is the number field, which is a finite degree field extension of the rational numbers \( \mathbb{Q} \). If \( \alpha \) is an algebraic number, then the smallest field that contains both \( \mathbb{Q} \) and \( \alpha \) is called \( \mathbb{Q}(\alpha) \). This field is a vector space over \( \mathbb{Q} \) with finite dimension, known as the degree of the extension.

Ring of Integers

Within number fields, we find the ring of integers, which generalizes the ring of ordinary integers \( \mathbb{Z} \). For a number field \( K \), its ring of integers \( \mathcal{O}_K \) includes all elements of \( K \) that are roots of monic polynomials with coefficients in \( \mathbb{Z} \). For instance, in the quadratic field \( \mathbb{Q}(\sqrt{d}) \), the ring of integers generally consists of numbers of the form \( a + b\sqrt{d} \) where \( a \) and \( b \) are integers.

Ideals and Factorization

A significant focus within algebraic number theory is the study of ideals in rings of integers. Unlike the ring of standard integers where every number has a unique prime factorization (the Fundamental Theorem of Arithmetic), unique factorization can fail in general rings. To navigate this, algebraic number theorists utilize the concept of ideals. A non-zero ideal in \( \mathcal{O}_K \) is a subset of the ring that absorbs multiplication by elements of the ring and satisfies certain additive properties.

Prime Ideals

In this context, an ideal \( \mathfrak{p} \) is called prime if whenever \( \mathfrak{p} \) divides a product of two ideals \( \mathfrak{a} \mathfrak{b} \), \( \mathfrak{p} \) must divide at least one of \( \mathfrak{a} \) or \( \mathfrak{b} \). The factorization of ideals into prime ideals restores a form of unique factorization in many cases.

Class Group

A central object of study is the ideal class group, which measures the failure of unique factorization in \( \mathcal{O}_K \). It is defined as the group of fractional ideals modulo the principal ideals, and its finite nature has profound implications for the structure of numbers in the given field. The size of the ideal class group is known as the class number.

Units and Dirichlet’s Unit Theorem

The group of units \( \mathcal{O}_K^* \) of \( \mathcal{O}_K \) (elements which have a multiplicative inverse in \( \mathcal{O}_K \)) also plays a significant role. Dirichlet’s Unit Theorem provides a description of this group for any number field \( K \), stating that \( \mathcal{O}_K^* \) is finitely generated and of the form:
\[ \mathcal{O}_K^* \cong \mu(K) \times \mathbb{Z}^r \]
where \( \mu(K) \) is the finite cyclic group of roots of unity in \( K \), and \( r \) is related to the number of real and complex embeddings of \( K \).

Summary

Algebraic number theory combines the elegance of algebra with the complexity of number theory to explore the deeper properties of numbers beyond the reach of elementary methods. By extending the concept of integers and examining the structures and symmetries within these extensions, it provides a rich and intricate framework that has been instrumental in solving many classical problems in mathematics.