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What (who) is field servoid - definition
![genus]] two (two handles). The genus can be read off the field of meromorphic functions on the surface.](https://commons.wikimedia.org/wiki/Special:FilePath/Double torus illustration.png?width=200 "genus]] two (two handles). The genus can be read off the field of meromorphic functions on the surface.")
![The fifth roots of unity form a [[regular pentagon]].](https://commons.wikimedia.org/wiki/Special:FilePath/One5Root.svg?width=200 "The fifth roots of unity form a [[regular pentagon]].")
![A field of [[rapeseed]]s in [[Kärkölä]], Finland (2010)](https://commons.wikimedia.org/wiki/Special:FilePath/Field in Kärkölä.jpg?width=200 "A field of [[rapeseed]]s in [[Kärkölä]], Finland (2010)")
![Spring fields with trees, [[Majorca]], Spain, 2004](https://commons.wikimedia.org/wiki/Special:FilePath/Spring fields Mallorca.jpg?width=200 "Spring fields with trees, [[Majorca]], Spain, 2004")
![A [[combine harvester]] on the field, [[Pornainen]], Finland, 2016](https://commons.wikimedia.org/wiki/Special:FilePath/2016 Pornainen 27.jpg?width=200 "A [[combine harvester]] on the field, [[Pornainen]], Finland, 2016")
![Sown fields in an [[open field system]] of farming](https://commons.wikimedia.org/wiki/Special:FilePath/Moforlogiaagrária.jpg?width=200 "Sown fields in an [[open field system]] of farming")
![[[Paddy field]]](https://commons.wikimedia.org/wiki/Special:FilePath/Paddy field of near Gingee Fort.jpg?width=200 "[[Paddy field]]")
![In [[classical gravitation]], mass is the source of an attractive [[gravitational field]] '''g'''.](https://commons.wikimedia.org/wiki/Special:FilePath/Newtonian gravity field (physics).svg?width=200 "In [[classical gravitation]], mass is the source of an attractive [[gravitational field]] '''g'''.")
![Fields due to [[color charge]]s, like in [[quark]]s ('''G''' is the [[gluon field strength tensor]]). These are "colorless" combinations. '''Top:''' Color charge has "ternary neutral states" as well as binary neutrality (analogous to [[electric charge]]). '''Bottom:''' The quark/antiquark combinations.<ref name="Mc Graw Hill"/><ref name="M. Mansfield, C. O’Sullivan 2011"/>](https://commons.wikimedia.org/wiki/Special:FilePath/Qcd fields field (physics).svg?width=200 "Fields due to [[color charge]]s, like in [[quark]]s ('''G''' is the [[gluon field strength tensor]]). These are \"colorless\" combinations. '''Top:''' Color charge has \"ternary neutral states\" as well as binary neutrality (analogous to [[electric charge]]). '''Bottom:''' The quark/antiquark combinations.<ref name=\"Mc Graw Hill\"/><ref name=\"M. Mansfield, C. O’Sullivan 2011\"/>")
.svg?width=200 "isbn=0-691-03323-4}}</ref>")
Wikipedia
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.
The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements.
The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that angle trisection and squaring the circle cannot be done with a compass and straightedge. Moreover, it shows that quintic equations are, in general, algebraically unsolvable.
Fields serve as foundational notions in several mathematical domains. This includes different branches of mathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Number fields, the siblings of the field of rational numbers, are studied in depth in number theory. Function fields can help describe properties of geometric objects.
