View source for Field

{{MSC|12}}
{{TEX|done}}

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|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|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|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|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|Galois theory]] deals with finite separable extensions
(cf.
[[Separable extension|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|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.

====Comments====
The German term for "field" is "Körper" and this is
of course the term used in
{{Cite|Le}}. The edition cited here as
{{Cite|Le}} is a corrected reprint of the 4th edition
(Braunschweig, 1893); the 1871 edition was the second.


====References====
{|
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|valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Eléments de mathematique. Algèbre", Masson (1981) pp. Chapt. 4–7 {{MR|0643362}} {{ZBL|0498.12001}}
|-
|valign="top"|{{Ref|Ja}}||valign="top"| N. Jacobson, "Lectures in abstract algebra", '''1. Basic concepts''', Springer (1975) {{MR|0041102}} {{ZBL|0326.00001}}
|-
|valign="top"|{{Ref|La}}||valign="top"| S. Lang, "Algebra", Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}}
|-
|valign="top"|{{Ref|Le}}||valign="top"| P.G. Lejeune-Dirichlet, "Zahlentheorie", Chelsea, reprint (1968) {{MR|}} {{ZBL|25.0252.01}} {{ZBL|03.0063.01}}
|-
|valign="top"|{{Ref|Wa}}||valign="top"| B.L. van der Waerden, "Algebra", '''1–2''', Springer (1967–1971) (Translated from German) {{MR|0263582}} {{ZBL|1032.00001}} {{ZBL|1032.00002}} 
|-
|valign="top"|{{Ref|ZaSa}}||valign="top"| O. Zariski, P. Samuel, "Commutative algebra", '''1''', Springer (1975) {{MR|0389876}} {{MR|0384768}} {{ZBL|0313.13001}}
|-
|}