Number field
Definition and Basic Properties
A number field is a field that is a finite extension of the field of rational numbers \(\mathbb{Q}\). In other words, a number field is a field that contains \(\mathbb{Q}\) and has finite degree over \(\mathbb{Q}\). The study of number fields is a central topic in algebraic number theory, which is a branch of number theory that deals with the properties of numbers in these fields.
Number fields can be classified by their degree, which is the dimension of the field as a vector space over \(\mathbb{Q}\). If \(K\) is a number field and \([K : \mathbb{Q}] = n\), then \(n\) is called the degree of the number field.
Examples of Number Fields
One of the simplest examples of a number field is the field of rational numbers itself, \(\mathbb{Q}\). More interesting examples include the quadratic fields, which are degree 2 extensions of \(\mathbb{Q}\). A quadratic field can be written in the form \(\mathbb{Q}(\sqrt{d})\), where \(d\) is a square-free integer.
Another important class of number fields are the cyclotomic fields, which are obtained by adjoining a primitive \(n\)-th root of unity to \(\mathbb{Q}\). These fields have applications in Galois theory and modular arithmetic.
Algebraic Integers and the Ring of Integers
In a number field \(K\), an element \(\alpha \in K\) is called an algebraic integer if it is a root of a monic polynomial with coefficients in \(\mathbb{Z}\). The set of all algebraic integers in \(K\) forms a ring, known as the ring of integers of \(K\), denoted by \(\mathcal{O}_K\).
The ring of integers \(\mathcal{O}_K\) is a Dedekind domain, which means it has several important properties, such as the existence of unique factorization of ideals into prime ideals. However, unlike \(\mathbb{Z}\), the ring of integers \(\mathcal{O}_K\) does not necessarily have unique factorization of elements into prime elements.
Ideal Theory in Number Fields
The study of ideals in the ring of integers \(\mathcal{O}_K\) is a fundamental aspect of algebraic number theory. An ideal in \(\mathcal{O}_K\) is a subset that is closed under addition and under multiplication by any element of \(\mathcal{O}_K\). Ideals can be classified into prime ideals and fractional ideals.
One of the key results in the theory of ideals is the unique factorization of ideals in \(\mathcal{O}_K\). This theorem states that every non-zero ideal in \(\mathcal{O}_K\) can be uniquely factored into a product of prime ideals. This property is crucial for defining and studying the class group of a number field.
Class Group and Class Number
The class group of a number field \(K\) is the group of fractional ideals of \(\mathcal{O}_K\) modulo the principal ideals. The class group measures the failure of unique factorization in \(\mathcal{O}_K\). The order of the class group, known as the class number, is a fundamental invariant of the number field.
The class number provides important information about the arithmetic of the number field. For example, a number field with class number one has unique factorization of elements in its ring of integers. The study of class numbers and class groups is a central topic in algebraic number theory.
Units and the Unit Group
The units of the ring of integers \(\mathcal{O}_K\) are the elements that have a multiplicative inverse in \(\mathcal{O}_K\). The set of units forms a group under multiplication, known as the unit group of \(\mathcal{O}_K\), denoted by \(\mathcal{O}_K^\times\).
The structure of the unit group is described by Dirichlet's unit theorem, which states that \(\mathcal{O}_K^\times\) is isomorphic to \(\mathbb{Z}^{r+s-1} \times \mu_K\), where \(r\) is the number of real embeddings of \(K\), \(s\) is the number of pairs of complex embeddings, and \(\mu_K\) is the group of roots of unity in \(K\).
Galois Theory and Number Fields
Galois theory provides a powerful framework for studying the symmetries of number fields. A number field \(K\) is called a Galois extension of \(\mathbb{Q}\) if it is the splitting field of a polynomial with coefficients in \(\mathbb{Q}\). The Galois group of \(K\) over \(\mathbb{Q}\) is the group of field automorphisms of \(K\) that fix \(\mathbb{Q}\).
The structure of the Galois group provides deep insights into the arithmetic properties of the number field. For example, the Kronecker-Weber theorem states that every abelian extension of \(\mathbb{Q}\) is contained in a cyclotomic field.
Local Fields and Completions
The study of number fields often involves considering their completions at various places. A local field is a field that is complete with respect to a discrete valuation and has a finite residue field. The most common examples are the fields of p-adic numbers \(\mathbb{Q}_p\) and the field of real numbers \(\mathbb{R}\).
For a number field \(K\), the completion at a place corresponding to a prime ideal \(\mathfrak{p}\) in \(\mathcal{O}_K\) is a local field, denoted by \(K_\mathfrak{p}\). The study of local fields and their properties is essential for understanding the global properties of number fields.
Applications of Number Fields
Number fields have numerous applications in various areas of mathematics. They play a crucial role in Diophantine equations, where solutions are sought in the ring of integers of a number field. They are also fundamental in the study of L-functions and modular forms, which are central objects in modern number theory.
Number fields are also used in cryptography, particularly in public-key cryptography schemes such as RSA and elliptic curve cryptography. The arithmetic of number fields provides the theoretical foundation for these cryptographic protocols.