Algebraic number theory

Introduction

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the properties of algebraic numbers. This field primarily focuses on algebraic integers, which are roots of monic polynomials with integer coefficients. Algebraic number theory has deep connections with various areas of mathematics, including field theory, Galois theory, and algebraic geometry.

Historical Background

The origins of algebraic number theory can be traced back to the work of mathematicians such as Carl Friedrich Gauss, who laid the groundwork in his book "Disquisitiones Arithmeticae" in 1801. The field was further developed by Ernst Kummer, Richard Dedekind, and Leopold Kronecker in the 19th century. Kummer introduced the concept of ideal numbers to address the failure of unique factorization in certain number fields. Dedekind later refined this idea into the modern concept of ideals.

Basic Concepts

Algebraic Numbers and Algebraic Integers

An algebraic number is any complex number that is a root of a non-zero polynomial equation with rational coefficients. If the polynomial can be taken to have integer coefficients and a leading coefficient of one, the root is called an algebraic integer. For example, the number \(\sqrt{2}\) is an algebraic integer because it is a root of the polynomial \(x^2 - 2 = 0\).

Number Fields

A number field is a finite degree field extension of the field of rational numbers \(\mathbb{Q}\). If \(K\) is a number field, then every element of \(K\) is an algebraic number. The degree of the field extension is called the degree of the number field. For example, \(\mathbb{Q}(\sqrt{2})\) is a number field of degree 2 over \(\mathbb{Q}\).

Rings of Integers

The ring of integers of a number field \(K\), denoted \(\mathcal{O}_K\), is the set of all algebraic integers in \(K\). This ring generalizes the concept of the integers \(\mathbb{Z}\) in \(\mathbb{Q}\). For example, the ring of integers in \(\mathbb{Q}(\sqrt{2})\) is \(\mathbb{Z}[\sqrt{2}]\).

Ideals and Factorization

Ideals

In algebraic number theory, an ideal is a generalization of the concept of a number being divisible by another number. An ideal in a ring \(\mathcal{O}_K\) is a subset of \(\mathcal{O}_K\) that is closed under addition and under multiplication by any element of \(\mathcal{O}_K\). Ideals can be added, multiplied, and intersected, forming a rich algebraic structure.

Prime Ideals and Unique Factorization

A prime ideal is an ideal that has properties similar to those of a prime number. In the ring of integers \(\mathcal{O}_K\), every ideal can be uniquely factored into a product of prime ideals. This is known as the unique factorization of ideals. This property is crucial for many proofs and theorems in algebraic number theory.

Class Field Theory

Class field theory is a major area of algebraic number theory that studies abelian extensions of number fields. It provides a deep understanding of the relationship between the arithmetic of a number field and its field extensions. The main theorems of class field theory describe the abelian extensions of a number field in terms of the field's ideals and units.

Galois Theory and Algebraic Number Theory

Galois theory, developed by Évariste Galois, is a powerful tool in algebraic number theory. It studies the symmetries of the roots of a polynomial equation. The Galois group of a polynomial is a group of permutations of the roots that preserves the algebraic relations among them. Galois theory provides a deep connection between field theory and group theory, and it is essential for understanding the structure of field extensions.

Diophantine Equations

Diophantine equations are polynomial equations where the solutions are required to be integers or rational numbers. Algebraic number theory provides tools for solving these equations by embedding them into number fields and studying their solutions in the context of algebraic integers and ideals. Famous examples include Fermat's Last Theorem and the Mordell-Weil theorem.

Applications

Algebraic number theory has numerous applications in various fields of mathematics and beyond. It plays a crucial role in cryptography, particularly in the construction of elliptic curve cryptography and RSA encryption. It is also fundamental in coding theory, mathematical physics, and algebraic geometry.

References

Gauss, Carl Friedrich. "Disquisitiones Arithmeticae."
Kummer, Ernst. "On the Theory of Ideal Numbers."
Dedekind, Richard. "Theory of Algebraic Integers."