algebraic structure with addition, multiplication and division
this article is about fields in algebra. for fields in geometry, see
vector field. for other uses, see
field § mathematics.
the
regular heptagon cannot be constructed using only a
straightedge and compass construction; this can be proven using the field of
constructible numbers.
algebraic structures 

 group
 semigroup / monoid
 rack and quandle
 quasigroup and loop
 abelian group
 magma
 lie group
group theory 
 ring
 semiring
 nearring
 commutative ring
 integral domain
 field
 division ring
ring theory 
 lattice
 semilattice
 complemented lattice
 total order
 heyting algebra
 boolean algebra
 map of lattices
 lattice theory

 module
 group with operators
 vector space

 associative
 nonassociative
 composition algebra
 lie algebra
 graded
 bialgebra


algebraic structure → ring theory ring theory 


basic concepts rings
 • subrings
 • ideal
 • quotient ring
 • fractional ideal
 • total quotient ring
 • product ring
 • free product of associative algebras
 • tensor product of rings
ring homomorphisms
 • kernel
 • inner automorphism
 • frobenius endomorphism
algebraic structures
 • module
 • associative algebra
 • graded ring
 • involutive ring
 • category of rings
 • initial ring $\mathbb{z}$
 • terminal ring $0={\mathbb{z}}_{1}$
related structures
 • field
 • finite field
 • nonassociative ring
 • lie ring
 • jordan ring
 • semiring
 • semifield


noncommutative rings
 • division ring
 • semiprimitive ring
 • simple ring
 • commutator
noncommutative algebraic geometry
free algebra
clifford algebra
 • geometric algebra
operator algebra 

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 padic 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 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.