In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an algebraic number field (or simply number field) is an
extension field of the
field of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s such that the
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
has
finite degree (and hence is an
algebraic
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
field extension).
Thus
is a field that contains
and has finite
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
when considered as a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over
The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic o ...
. This study reveals hidden structures behind usual 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
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of elements together with two operations, namely
addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
, and
multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
, and some distributivity assumptions. A prominent example of a field is the field of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s, commonly denoted together with its usual operations of addition and multiplication.
Another notion needed to define algebraic number fields is
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s. To the extent needed here, vector spaces can be thought of as consisting of sequences (or
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
s)
:(''x''
1, ''x''
2, …)
whose entries are elements of a fixed field, such as the field Any two such sequences can be added by adding the entries one per one. Furthermore, 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 are of infinite length. If, however, the vector space consists of ''finite'' sequences
:(''x''
1, ''x''
2, …, ''x''
''n''),
the vector space is said to be of finite
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
, ''n''.
Definition
An algebraic number field (or simply number field) is a finite-
degree field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
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, in general.
* The
Gaussian rationals, denoted
(read as "
adjoined "), form the first (historically) non-trivial example of a number field. Its elements are elements of the form
where both ''a'' and ''b'' are rational numbers and ''i'' is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
. Such expressions may be added, subtracted, and multiplied according to the usual rules of arithmetic and then simplified using the identity
Explicitly,
Non-zero Gaussian rational numbers are
invertible, which can be seen from the identity
It follows that the Gaussian rationals form a number field which is two-dimensional as a vector space over
* 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
th root of unity
. This field contains all complex ''n''th roots of unity and its dimension over
is equal to
, where
is the
Euler totient function.
Non-Examples
* The
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, and the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, are fields which 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.
* The set
of
ordered pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s 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 divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...
s:
Algebraicity, and ring of integers
Generally, in
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, a field extension
is
algebraic
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
if every element
of the bigger field
is the zero of a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
with coefficients
in
:
Every field extension of finite degree is algebraic. (Proof: for
in simply consider
– we get a linear dependence, i.e. a polynomial that
is a root of.) 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 algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form:
:x^n+c_x^+\ ...
. In general it will have rational coefficients.
If, however, its coefficients are actually all integers,
is called an ''
algebraic integer''.
Any (usual) 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
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
of (that is, a ring contained in) 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
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
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'' (or ''Dedekind domain''), in honor of
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
, 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 ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
s. For example, the ideal
in the ring