Editing Field (mathematics) (section)

== History ==
Historically, three algebraic disciplines led to the concept of a field: the question of solving polynomial equations, [[algebraic number theory]], and [[algebraic geometry]].<ref>{{harvp|Kleiner|2007|loc=p. 63}}</ref> A first step towards the notion of a field was made in 1770 by [[Joseph-Louis Lagrange]], who observed that permuting the zeros {{math|''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>}} of a [[cubic polynomial]] in the expression
: {{math|(''x''<sub>1</sub> + ''ωx''<sub>2</sub> + ''ω''<sup>2</sup>''x''<sub>3</sub>)<sup>3</sup>}}
(with {{math|''ω''}} being a third [[root of unity]]) only yields two values. This way, Lagrange conceptually explained the classical solution method of [[Scipione del Ferro]] and [[François Viète]], which proceeds by reducing a cubic equation for an unknown {{math|''x''}} to a quadratic equation for {{math|''x''<sup>3</sup>}}.<ref>{{harvp|Kiernan|1971|loc=p. 50}}</ref> Together with a similar observation for [[quartic polynomial|equations of degree 4]], Lagrange thus linked what eventually became the concept of fields and the concept of groups.<ref>{{harvp|Bourbaki|1994|loc=pp. 75–76}}</ref> [[Alexandre-Théophile Vandermonde|Vandermonde]], also in 1770, and to a fuller extent, [[Carl Friedrich Gauss]], in his ''[[Disquisitiones Arithmeticae]]'' (1801), studied the equation
: {{math|1=''x''<sup>&hairsp;''p''</sup> = 1}}
for a prime {{math|''p''}} and, again using modern language, the resulting cyclic [[Galois group]]. Gauss deduced that a [[regular polygon|regular {{math|''p''}}-gon]] can be constructed if {{math|1=''p'' = 2<sup>2<sup>''k''</sup></sup> + 1}}. Building on Lagrange's work, [[Paolo Ruffini (mathematician)|Paolo Ruffini]] claimed (1799) that [[quintic equation]]s (polynomial equations of degree {{math|5}}) cannot be solved algebraically; however, his arguments were flawed. These gaps were filled by [[Niels Henrik Abel]] in 1824.<ref>{{harvp|Corry|2004|loc=p. 24}}</ref> [[Évariste Galois]], in 1832, devised necessary and sufficient criteria for a polynomial equation to be algebraically solvable, thus establishing in effect what is known as [[Galois theory]] today. Both Abel and Galois worked with what is today called an [[algebraic number field]], but conceived neither an explicit notion of a field, nor of a group.

In 1871 [[Richard Dedekind]] introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, the [[German (language)|German]] word ''Körper'', which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by {{harvp|Moore|1893}}.<ref>{{cite web| url = http://jeff560.tripod.com/f.html| title = ''Earliest Known Uses of Some of the Words of Mathematics (F)''}}</ref>

{{Blockquote|text=By a field we will mean every infinite system of real or complex numbers so closed in itself and perfect that addition, subtraction, multiplication, and division of any two of these numbers again yields a number of the system.
|author=Richard Dedekind, 1871<ref>{{harvp|Dirichlet|1871|loc=p. 42}}, translation by {{harvp|Kleiner|2007|loc=p. 66}}</ref>}}

In 1881 [[Leopold Kronecker]] defined what he called a ''domain of rationality'', which is a field of [[rational fraction]]s in modern terms. Kronecker's notion did not cover the field of all algebraic numbers (which is a field in Dedekind's sense), but on the other hand was more abstract than Dedekind's in that it made no specific assumption on the nature of the elements of a field. Kronecker interpreted a field such as {{math|'''Q'''(π)}} abstractly as the rational function field {{math|'''Q'''(''X'')}}. Prior to this, examples of transcendental numbers were known since [[Joseph Liouville]]'s work in 1844, until [[Charles Hermite]] (1873) and [[Ferdinand von Lindemann]] (1882) proved the transcendence of {{math|''e''}} and {{math|''π''}}, respectively.<ref>{{harvp|Bourbaki|1994|loc=p. 81}}</ref>

The first clear definition of an abstract field is due to {{harvp|Weber|1893}}.<ref>{{harvp|Corry|2004|loc=p. 33}}. See also {{harvp|Fricke|Weber|1924}}.</ref> In particular, [[Heinrich Martin Weber]]'s notion included the field {{math|'''F'''<sub>''p''</sub>}}. [[Giuseppe Veronese]] (1891) studied the field of formal power series, which led {{harvp|Hensel|1904}} to introduce the field of {{math|''p''}}-adic numbers. {{harvp|Steinitz|1910}} synthesized the knowledge of abstract field theory accumulated so far. He axiomatically studied the properties of fields and defined many important field-theoretic concepts. The majority of the theorems mentioned in the sections [[#Galois theory|Galois theory]], [[#Constructing fields|Constructing fields]] and [[#Elementary notions|Elementary notions]] can be found in Steinitz's work. {{harvp|Artin|Schreier|1927}} linked the notion of [[ordered field|orderings in a field]], and thus the area of analysis, to purely algebraic properties.<ref>{{harvp|Bourbaki|1994|loc=p. 92}}</ref> [[Emil Artin]] redeveloped Galois theory from 1928 through 1942, eliminating the dependency on the [[primitive element theorem]].