          # 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.

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 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.

• definition
• examples
• elementary notions
• finite fields
• history
• constructing fields
## 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-like Group Semigroup / Monoid Rack and quandle Quasigroup and loop Abelian group Magma Lie group Group theory Ring-like Ring Semiring Near-ring Commutative ring Integral domain Field Division ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map of lattices Lattice theory Module-like Module Group with operators Vector space Linear algebra Algebra-like Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra vt Algebraic structure → Ring theoryRing theory Basic conceptsRings • 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 • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield Commutative algebraCommutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Composition ring • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number field • Ring of integers • Algebraic independence • Transcendental number theory • Transcendence degree p-adic number theory and decimals • Direct limit/Inverse limit • Zero ring $\mathbb {Z} _{1}$ • Integers modulo pn $\mathbb {Z} /p^{n}\mathbb {Z}$ • Prüfer p-ring $\mathbb {Z} (p^{\infty })$ • Base-p circle ring $\mathbb {T}$ • Base-p integers $\mathbb {Z}$ • p-adic rationals $\mathbb {Z} [1/p]$ • Base-p real numbers $\mathbb {R}$ • p-adic integers $\mathbb {Z} _{p}$ • p-adic numbers $\mathbb {Q} _{p}$ • p-adic solenoid $\mathbb {T} _{p}$ Algebraic geometry • Affine variety Noncommutative algebraNoncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator Noncommutative algebraic geometry Free algebra Clifford algebra • Geometric algebra Operator algebra v 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 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. Contents 1 Definition 1.1 Classic definition 1.2 Alternative definition 2 Examples 2.1 Rational numbers 2.2 Real and complex numbers 2.3 Constructible numbers 2.4 A field with four elements 3 Elementary notions 3.1 Consequences of the definition 3.2 The additive and the multiplicative group of a field 3.3 Characteristic 3.4 Subfields and prime fields 4 Finite fields 5 History 6 Constructing fields 6.1 Constructing fields from rings 6.1.1 Field of fractions 6.1.2 Residue fields 6.2 Constructing fields within a bigger field 6.3 Field extensions 6.3.1 Algebraic extensions 6.3.2 Transcendence bases 6.4 Closure operations 7 Fields with additional structure 7.1 Ordered fields 7.2 Topological fields 7.2.1 Local fields 7.3 Differential fields 8 Galois theory 9 Invariants of fields 9.1 Model theory of fields 9.2 The absolute Galois group 9.3 K-theory 10 Applications 10.1 Linear algebra and commutative algebra 10.2 Finite fields: cryptography and coding theory 10.3 Geometry: field of functions 10.4 Number theory: global fields 11 Related notions 11.1 Division rings 12 Notes 13 References  