Editing Field (mathematics) (section)

{{Short description|Algebraic structure with addition, multiplication, and division}}
{{About|a commutative algebraic structure|the non-commutative generalization|Skew field|vector valued functions|Vector field|other uses|Field (disambiguation)#Mathematics}}
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[[File:Regular_polygon_7_annotated.svg|thumb|262px|The [[regular polygon|regular]] [[heptagon]] cannot be constructed using only a [[straightedge and compass construction]]; this can be proven using the field of [[constructible number]]s.]]
{{Algebraic structures}}
In [[mathematics]], a '''field''' is a [[set (mathematics)|set]] on which [[addition]], [[subtraction]], [[multiplication]], and [[division (mathematics)|division]] are defined and behave as the corresponding operations on [[rational number|rational]] and [[real number]]s. 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 number]]s, the field of [[real number]]s and the field of [[complex number]]s. Many other fields, such as [[field of rational functions|fields of rational functions]], [[algebraic function field]]s, [[algebraic number field]]s, and [[p-adic number|''p''-adic fields]] are commonly used and studied in mathematics, particularly in number theory and [[algebraic geometry]]. Most [[cryptographic protocol]]s rely on [[finite field]]s, i.e., fields with finitely many [[element (set)|elements]].

The theory of fields proves that [[angle trisection]] and [[squaring the circle]] cannot be done with a [[compass and straightedge]]. [[Galois theory]], devoted to understanding the symmetries of [[field extension]]s, provides an elegant proof of the [[Abel–Ruffini theorem]] that general [[quintic equation]]s cannot be [[solution in radicals|solved in radicals]].

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 [[scalar (mathematics)|scalars]] for a [[vector space]], which is the standard general context for [[linear algebra]]. [[Number field]]s, the siblings of the field of rational numbers, are studied in depth in [[number theory]]. [[Function field of an algebraic variety|Function fields]] can help describe properties of geometric objects.