Algebraic number field


In mathematics, an algebraic number field is an extension field of the field of rational numbers such that the field extension has finite degree.
Thus is a field that contains and has finite dimension when considered as a vector space over
The study of algebraic number fields, that is, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind the rational numbers, by using algebraic methods.

Definition

Prerequisites

The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. These operations make the field into an abelian group under addition, and they make the nonzero elements of the field into another abelian group under multiplication. A prominent example of a field is the field of rational numbers, commonly denoted together with its usual operations of addition and multiplication.
Another notion needed to define algebraic number fields is vector spaces. To the extent needed here, vector spaces can be thought of as consisting of sequences
whose entries are elements of a fixed field, such as the field Any two such sequences can be added by adding the corresponding entries. Furthermore, all members of any sequence can be multiplied by a single element c of the fixed field. These two operations known as vector addition and scalar multiplication satisfy a number of properties that serve to define vector spaces abstractly. Vector spaces are allowed to be "infinite-dimensional", that is to say that the sequences constituting the vector spaces may be of infinite length. If, however, the vector space consists of finite sequences
the vector space is said to be of finite dimension,.

Definition

An algebraic number field is a finite-degree field extension of the field of rational numbers. Here degree means the dimension of the field as a vector space over

Examples

  • The smallest and most basic number field is the field of rational numbers. Many properties of general number fields are modeled after the properties of. At the same time, many other properties of algebraic number fields are substantially different from the properties of rational numbers—one notable example is that the ring of algebraic integers of a number field is not a principal ideal domain, and not even a unique factorization domain, in general.
  • The Gaussian rationals, denoted , form the first non-trivial example of a number field. Its elements are elements of the form where both and are rational numbers and is the imaginary unit. Such expressions may be added, subtracted, and multiplied according to the usual rules of arithmetic and then simplified using the identity. Explicitly, for real numbers :
  • More generally, for any square-free integer, the quadratic field is a number field obtained by adjoining the square root of to the field of rational numbers. Arithmetic operations in this field are defined in analogy with the case of Gaussian rational numbers,.
  • The cyclotomic field where, is a number field obtained from by adjoining a primitive n-th root of unity. This field contains all complex nth roots of unity and its dimension over is equal to, where is the Euler totient function.

    Non-examples

  • The real numbers, and the complex numbers, are fields that have infinite dimension as -vector spaces; hence, they are not number fields. This follows from the uncountability of and as sets, whereas every number field is necessarily countable, as they are finite-dimensional vector spaces over.
  • The set of ordered pairs of rational numbers, with the entry-wise addition and multiplication is a two-dimensional commutative algebra over However, it is not a field, since it has zero divisors:.

    Algebraicity, and ring of integers

Generally, in abstract algebra, a field extension is algebraic if every element of the bigger field is the zero of a polynomial with coefficients in
Every field extension of finite degree is algebraic. In particular this applies to algebraic number fields, so any element of an algebraic number field can be written as a zero of a polynomial with rational coefficients. Therefore, elements of are also referred to as algebraic numbers. Given a polynomial such that, it can be arranged such that the leading coefficient is one, by dividing all coefficients by it, if necessary. A polynomial with this property is known as a monic polynomial. In general it will have rational coefficients.
If, however, the monic polynomial's coefficients are actually all integers, is called an algebraic integer.
Any integer is an algebraic integer, as it is the zero of the linear monic polynomial:
It can be shown that any algebraic integer that is also a rational number must actually be an integer, hence the name "algebraic integer". Again using abstract algebra, specifically the notion of a finitely generated module, it can be shown that the sum and the product of any two algebraic integers is still an algebraic integer. It follows that the algebraic integers in form a ring denoted called the ring of integers of It is a subring of A field contains no zero divisors and this property is inherited by any subring, so the ring of integers of is an integral domain. The field is the field of fractions of the integral domain This way one can get back and forth between the algebraic number field and its ring of integers Rings of algebraic integers have three distinctive properties: firstly, is an integral domain that is integrally closed in its field of fractions Secondly, is a Noetherian ring. Finally, every nonzero prime ideal of is maximal or, equivalently, the Krull dimension of this ring is one. An abstract commutative ring with these three properties is called a Dedekind ring, in honor of Richard Dedekind, who undertook a deep study of rings of algebraic integers.

Unique factorization

For general Dedekind rings, in particular rings of integers, there is a unique factorization of ideals into a product of prime ideals. For example, the ideal in the ring of quadratic integers factors into prime ideals as
However, unlike as the ring of integers of the ring of integers of a proper extension of need not admit unique factorization of numbers into a product of prime numbers or, more precisely, prime elements. This happens already for quadratic integers, for example in the uniqueness of the factorization fails:
Using the norm it can be shown that these two factorization are actually inequivalent in the sense that the factors do not just differ by a unit in Euclidean domains are unique factorization domains: For example the ring of Gaussian integers, and the ring of Eisenstein integers, where is a cube root of unity, have this property.

Analytic objects: ζ-functions, ''L''-functions, and class number formula

The failure of unique factorization is measured by the class number, commonly denoted h, the cardinality of the so-called ideal class group. This group is always finite. The ring of integers possesses unique factorization if and only if it is a principal ring or, equivalently, if has class number 1. Given a number field, the class number is often difficult to compute. The class number problem, going back to Gauss, is concerned with the existence of imaginary quadratic number fields with prescribed class number. The class number formula relates h to other fundamental invariants of It involves the Dedekind zeta function, a function in a complex variable, defined by
number of elements in the residue field The infinite product converges only for Re.
The Dedekind zeta-function generalizes the Riemann zeta-function in that ζ = ζ.
The class number formula states that ζ has a simple pole at s = 1 and at this point the residue is given by
Here r1 and r2 classically denote the number of real embeddings and pairs of complex embeddings of respectively. Moreover, Reg is the regulator of w the number of roots of unity in and D is the discriminant of
Dirichlet L-functions are a more refined variant of. Both types of functions encode the arithmetic behavior of and, respectively. For example, Dirichlet's theorem asserts that in any arithmetic progression
with coprime and, there are infinitely many prime numbers. This theorem is implied by the fact that the Dirichlet -function is nonzero at. Using much more advanced techniques including algebraic K-theory and Tamagawa measures, modern number theory deals with a description, if largely conjectural, of values of more general L-functions.

Bases for number fields

Integral basis

An integral basis for a number field of degree is a set
of n algebraic integers in such that every element of the ring of integers of can be written uniquely as a Z-linear combination of elements of B; that is, for any x in we have
where the mi are integers. It is then also the case that any element of can be written uniquely as
where now the mi are rational numbers. The algebraic integers of are then precisely those elements of where the mi are all integers.
Working locally and using tools such as the Frobenius map, it is always possible to explicitly compute such a basis, and it is now standard for computer algebra systems to have built-in programs to do this.

Power basis

Let be a number field of degree Among all possible bases of , there are particular ones known as power bases, that are bases of the form
for some element By the primitive element theorem, there exists such an, called a primitive element. If can be chosen in and such that is a basis of as a free Z-module, then is called a power integral basis, and the field is called a monogenic field. An example of a number field that is not monogenic was first given by Dedekind. His example is the field obtained by adjoining a root of the polynomial