Field

2010 Mathematics Subject Classification: Primary: 12-XX [MSN][ZBL]

A field is a commutative, associative ring containing a unit in which the set of non-zero elements is not empty and forms a group under multiplication (cf. Associative rings and algebras). A field may also be characterized as a simple non-zero commutative, associative ring containing a unit. Examples of fields: the field of rational numbers $\Q$, the field of real numbers $\R$, the field of complex numbers $\C$, finite fields (see Galois field), and the field of fractions of an integral domain.

A subfield of a field $K$ is a subset $M\subset K$ which itself is a field under the operations of addition and multiplication defined in $K$. For example, if $\def\s{\sigma} \s$ is some automorphism of a field $K$, then the set $$K^\s = \{ x\in K:\s(x)=x \}$$ is a subfield in $K$. If $M$ and $N$ are subfields of a field $K$, then their intersection $M\cap N$ is a subfield in $K$; also, there exists a smallest subfield $MN$ in the field $K$ that contains $M$ and $N$, called the composite of the fields $M$ and $N$ (in $K$). Each field contains a unique prime subfield (i.e. one not containing proper subfields).

Any field homomorphism is an imbedding. For an arbitrary field $K$ there exists a unique homomorphism $\def\phi{\varphi} \phi: \Z \to K$ which maps the unit of the ring $\Z$ to the unit of the field $K$. If $\ker \phi = 0$, $K$ is called a field of characteristic zero. In that case, the prime subfield of $K$ coincides with the field of fractions of the ring $\phi(\Z)$ and is isomorphic to the field $\Q$. If $\ker \phi \ne 0$, then $\ker\phi = p\Z$ for a certain prime $p$. This $p$ is called the characteristic of the field $K$. The prime subfield of $K$ coincides in that case with $\phi(\Z) \cong \Z/p\Z$.

If $k$ is a subfield of a field $K$, $K$ is called an extension of the field $k$. Let $Y$ be some subset in $K$. Then the field $k(Y)$ is defined as the smallest subfield of $K$ that contains $Y$ and $k$. It is said that $k(Y)$ is obtained from $k$ by adjoining the elements from the set $Y$.

Basic problems in the theory of fields consist of giving a description of all subfields of a given field, of all fields containing a given field, i.e. overfields (see Extension of a field), to examine all imbeddings of a field in some other field, to classify fields up to an isomorphism, and to examine the automorphism group of a given field.

A field $K$ is said to be finitely generated over a subfield $k$ if there exists a finite set $Y\subset K$ such that $K=k(Y)$. Any such field can be interpreted as the field of rational functions, $k(X)$, of a certain irreducible algebraic variety $X$ defined over $k$. Algebraic geometry deals, among other things, with the study of such fields. In particular, the classification of such fields is equivalent to the birational classification of irreducible algebraic varieties, and the problem of finding the group of automorphisms of a field $K=k(X)$ that leave all elements of the field $k$ invariant is equivalent to finding all birational automorphisms of the variety $X$ defined over $k$. Galois theory deals with finite separable extensions (cf. Separable extension) of arbitrary fields. An important part in number theory is played by the finite extensions of the field $\Q$, which are called algebraic number fields. Algebraic number theory deals with these fields.

Field theory also deals with fields having certain additional structures, such as differential fields, topological fields, ordered fields, formally real and formally $p$-adic fields, etc.

Field theory originated (within the framework of the theory of algebraic equations) in the middle of the 19th century. Papers by E. Galois and J.L. Lagrange on group theory and by C.F. Gauss on number theory made it clear that one had to examine the nature of number systems themselves. The concept of a field was put forward in papers by L. Kronecker and R. Dedekind. Dedekind introduced the concept of a field, which he originally called a "rational domainrational domain" . Dedekind's theory was published in the comments and supplements to P.G. Lejeune-Dirichlet's Zahlentheorie. In them, Dedekind substantially supplemented and extended the theory of numbers, the theory of ideals and the theory of finite fields. The term "field" first appeared in the edition of this book in 1871.