Algebraic number field

In mathematics, an algebraic number field (or simply number field) is an extension field ${\displaystyle K}$ of the field of rational numbers ${\displaystyle \mathbb {Q} }$ such that the field extension ${\displaystyle K/\mathbb {Q} }$ has finite degree (and hence is an algebraic field extension). Thus ${\displaystyle K}$ is a field that contains ${\displaystyle \mathbb {Q} }$ and has finite dimension when considered as a vector space over ${\displaystyle \mathbb {Q} }$.

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.

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 ${\displaystyle \mathbb {Q} }$, 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 (or tuples)

whose entries are elements of a fixed field, such as the field ${\displaystyle \mathbb {Q} }$. 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, ${\displaystyle n}$.

An algebraic number field (or simply 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 ${\displaystyle \mathbb {Q} }$.

Generally, in abstract algebra, a field extension ${\displaystyle K/L}$ is algebraic if every element ${\displaystyle f}$ of the bigger field ${\displaystyle K}$ is the zero of a (nonzero) polynomial with coefficients ${\displaystyle e_{0},\ldots ,e_{m}}$ in ${\displaystyle L}$: