1 Definition and basic properties
A field is an algebraic structure in which addition, subtraction, multiplication, and division can be carried out subject to familiar arithmetic rules. The concept abstracts the behavior of number systems such as the rationals, reals, and complex numbers. Fields are central in algebra because they provide a setting in which equations can be manipulated reliably, polynomial methods can be developed, and vector spaces can be defined.
1.1 Binary operations
A field has two binary operations: addition and multiplication. Each operation combines two elements of the set to produce another element of the same set. Addition is used to form sums, while multiplication is used to form products. These operations are required to interact in a controlled way so that arithmetic behaves predictably.
1.2 Axioms of a field
The defining axioms specify how addition and multiplication must behave. Together, they ensure that the structure supports ordinary algebraic manipulation.
1.2.1 Closure
The set must be closed under addition and multiplication, meaning that the sum or product of any two field elements is again a field element. This prevents arithmetic from leaving the system.
1.2.2 Associativity and commutativity
Addition and multiplication are each associative, so regrouping terms does not change the result. They are also commutative, so the order of terms does not matter. These properties make expressions easy to rearrange.
1.2.3 Identity elements
There must be an additive identity, usually written 0, such that adding it changes nothing. There must also be a multiplicative identity, usually written 1, such that multiplying by it leaves elements unchanged. The multiplicative identity is distinct from the additive identity.
1.2.4 Inverses
Every element has an additive inverse, which cancels it under addition. Every nonzero element also has a multiplicative inverse, which produces 1 when multiplied by the original element. This is what permits division by nonzero elements.
1.2.5 Distributive law
Multiplication distributes over addition. This means a product with a sum can be expanded into a sum of products. The distributive law links the two operations and underlies polynomial arithmetic.
1.3 Derived properties
Several useful facts follow from the axioms rather than being stated separately. These consequences are fundamental in calculations and proofs.
1.3.1 Uniqueness of identities and inverses
The additive identity and multiplicative identity are unique when they exist. Likewise, each element has exactly one additive inverse, and each nonzero element has exactly one multiplicative inverse. This uniqueness makes notation unambiguous.
1.3.2 Cancellation laws
If a common term is added to both sides of an equation, it can be canceled. The same is true for multiplication by a nonzero factor. These cancellation laws are essential for solving equations and simplifying expressions.
1.3.3 Division and subtraction
Subtraction is defined by adding an additive inverse, and division by a nonzero element is defined by multiplying by its multiplicative inverse. Thus, every field supports the usual arithmetic operations except division by zero.
2 Examples and non-examples
Fields arise naturally in many familiar number systems, but not every algebraic system qualifies. The difference often comes down to whether multiplicative inverses exist for all nonzero elements.
2.1 Standard fields
The most familiar fields are the number systems used in elementary and advanced mathematics.
2.1.1 Rational numbers
The rational numbers form a field under ordinary addition and multiplication. Every nonzero rational number has a rational reciprocal, so division by nonzero rationals stays within the system.
2.1.2 Real numbers
The real numbers also form a field. They extend the rationals by including limits of convergent sequences and serve as the foundation of classical analysis.
2.1.3 Complex numbers
The complex numbers form a field in which every nonzero element has a reciprocal. They are especially important in polynomial theory and in many areas of modern mathematics because algebraic equations often have solutions there even when they lack real ones.
2.2 Finite fields
Finite fields contain only finitely many elements. They play a major role in algebra, coding theory, and combinatorics because their arithmetic is both rigid and highly structured.
2.2.1 Prime fields
For each prime number p, there is a field with p elements. Its arithmetic is the same as modular arithmetic modulo p, and every nonzero element has a multiplicative inverse.
2.2.2 Fields of prime power order
For each prime power p^n, there exists a finite field with p^n elements, and any finite field has this form. These fields are built using polynomial constructions and are unique up to isomorphism for a given size.
2.3 Non-examples
Many familiar algebraic systems fail to be fields because they lack multiplicative inverses for some nonzero elements or fail another field axiom.
2.3.1 Integers
The integers are not a field because most nonzero integers do not have integer reciprocals. Although they form a ring, division generally leaves the set.
2.3.2 Matrix rings
Square matrices over a field do not form a field under ordinary matrix addition and multiplication, because nonzero matrices can fail to be invertible. They are noncommutative and contain zero divisors in the broader ring context.
2.3.3 Modular arithmetic with composite modulus
Arithmetic modulo a composite number is not a field because some nonzero residue classes do not have inverses. This is a common source of non-examples and contrasts with the prime modulus case.
3 Subfields and extensions
Fields can be related to one another by inclusion and enlargement. These relationships make it possible to study smaller systems inside larger ones and to build new fields by adjoining elements.
3.1 Subfields
A subfield is a subset of a field that is itself a field under the inherited operations. It must contain the same additive and multiplicative identities as the larger field.
3.1.1 Generated subfields
Given a set of elements in a field, the smallest subfield containing them is called the subfield they generate. It is formed by closing the set under addition, multiplication, subtraction, and division by nonzero elements.
3.2 Field extensions
A field extension is a larger field containing a given field as a subfield. Extensions allow new elements to be introduced while preserving the field structure.
3.2.1 Algebraic extensions
An element is algebraic over a field if it satisfies a nonzero polynomial with coefficients from that field. Extensions built by adjoining algebraic elements are called algebraic extensions.
3.2.2 Transcendental extensions
An element that satisfies no such polynomial is transcendental. Adjoining transcendental elements produces extensions that behave like fields of rational functions.
3.3 Degree of an extension
The degree of a field extension measures the size of the larger field as a vector space over the smaller one. It is one of the main numerical invariants in extension theory.
3.3.1 Finite extensions
An extension has finite degree when the larger field has a finite basis over the smaller field. Finite extensions are especially tractable and often arise from adjoining algebraic elements.
3.3.2 Tower law
If one field lies inside a second, which lies inside a third, the degrees multiply in the expected way when all are finite. This result is known as the tower law and is used repeatedly in extension arguments.
4 Special classes of fields
Certain fields are distinguished by additional properties that shape their arithmetic and geometry. These classes are important in classification and in the study of solvability of equations.
4.1 Characteristic
The characteristic of a field is the smallest positive number of times the multiplicative identity must be added to itself to obtain zero, if such a number exists. If no such number exists, the characteristic is zero.
4.1.1 Characteristic zero
Fields of characteristic zero include the rationals, reals, and complex numbers. In these fields, repeated addition of 1 never returns to 0.
4.1.2 Positive characteristic
In positive characteristic, repeated addition of 1 eventually yields 0. Finite fields always have positive characteristic, and this feature strongly influences their algebraic behavior.
4.2 Ordered fields
An ordered field is a field equipped with a compatible notion of order. The order must interact properly with addition and multiplication so that positive and negative elements behave as expected.
4.2.1 Formally real fields
A formally real field is one in which -1 cannot be expressed as a sum of squares. This condition is closely related to the possibility of ordering the field.
4.3 Algebraically closed fields
A field is algebraically closed if every nonconstant polynomial with coefficients in the field has a root in the field. Such fields are, in a sense, complete with respect to polynomial equations.
4.3.1 Closure properties
Algebraic closure means that polynomial factorization can always proceed as far as possible into linear factors. The complex numbers are the standard example, and many algebraic arguments become simpler in an algebraically closed setting.
4.4 Perfect fields
A perfect field is one in which every algebraic extension behaves especially well with respect to repeated roots. The notion is most relevant in positive characteristic.
4.4.1 Separable extensions
A separable extension is one in which elements are roots of polynomials without repeated roots, or equivalently polynomials with distinct algebraic behavior. Perfect fields are characterized by the fact that all algebraic extensions are separable.
5 Field homomorphisms and isomorphisms
Maps between fields preserve arithmetic and reveal structural similarities. They are indispensable in studying how fields relate to one another.
5.1 Field homomorphisms
A field homomorphism is a function between fields that preserves addition, multiplication, and the identities. Such maps send algebraic expressions in one field to corresponding expressions in another.
5.1.1 Kernels and images
The kernel of a field homomorphism consists of elements that map to zero, while the image is the set of values attained. For fields, a nontrivial homomorphism is highly constrained because kernels must respect the lack of nontrivial ideals.
5.2 Field isomorphisms
An isomorphism is a bijective field homomorphism. It shows that two fields are structurally identical, even if their elements are presented differently.
5.2.1 Automorphisms
An automorphism is an isomorphism from a field to itself. The collection of automorphisms of a field often reflects deep arithmetic structure and symmetry.
5.3 Fixed fields
Given a collection of automorphisms, the fixed field is the set of elements unchanged by all of them. Fixed fields encode the part of the field that is invariant under symmetry.
5.3.1 Galois groups
The Galois group of a field extension is the group of automorphisms of the larger field that fix the smaller field. It organizes the symmetry of algebraic extensions and connects fields with group theory.
6 Polynomials over fields
Polynomials are among the most important objects studied over fields. Field properties make polynomial arithmetic especially powerful and lead to major results about roots and factorization.
6.1 Polynomial rings
A polynomial ring consists of polynomials with coefficients in a field, using the usual addition and multiplication of polynomials. Its structure supports systematic algebraic manipulation.
6.1.1 Coefficients in a field
When coefficients lie in a field, division by nonzero coefficients is available, which simplifies many arguments. This setting is essential for standard results in factorization and root counting.
6.2 Roots and factorization
A root of a polynomial is a field element that makes the polynomial equal to zero. Factorization links roots to linear factors and helps classify polynomial behavior.
6.2.1 Fundamental theorem of algebra
The fundamental theorem of algebra states that every nonconstant polynomial with complex coefficients has at least one complex root. As a consequence, complex polynomials split completely into linear factors over the complex numbers.
6.3 Irreducible polynomials
An irreducible polynomial is one that cannot be factored into nonconstant polynomials over the same field. Such polynomials are the building blocks of polynomial factorization.
6.3.1 Minimal polynomials
The minimal polynomial of an algebraic element over a field is the unique monic irreducible polynomial of least degree having that element as a root. It captures the algebraic dependence of the element on the field.
7 Applications
Fields are used throughout mathematics because they provide a clean framework for handling linearity, solvability, and algebraic structure. Their applications range from geometry to number theory.
7.1 Linear algebra
Linear algebra is naturally formulated over a field, since vector spaces require scalar multiplication by field elements. Many basic results depend on the field axioms.
7.1.1 Vector spaces
A vector space is built from vectors that can be added together and scaled by field elements. The choice of field determines the kind of coordinates and numerical operations available.
7.1.2 Linear transformations
Linear transformations preserve vector addition and scalar multiplication. Their study depends on the underlying field and includes matrix representation, rank, and eigenvalue theory.
7.2 Number theory
Fields appear in number theory through modular arithmetic, polynomial congruences, and algebraic constructions. They provide a way to analyze integers using more flexible algebraic tools.
7.2.1 Modular arithmetic
When the modulus is prime, modular arithmetic forms a finite field. This allows calculations to be carried out with predictable inversion and factorization properties.
7.2.2 Diophantine equations
Field methods can assist in studying polynomial equations with integer or rational solutions. By enlarging the number system, one often gains insight into whether solutions exist and how they are structured.
7.3 Algebraic geometry
Algebraic geometry studies solution sets of polynomial equations using field-based algebraic tools. The field of coefficients strongly influences the shape and arithmetic of the resulting objects.
7.3.1 Coordinate rings
A coordinate ring records polynomial functions on an algebraic set and is built from a chosen field. It translates geometric questions into algebraic ones.
7.3.2 Rational points
Rational points are solutions whose coordinates lie in the chosen base field. Their study connects geometric objects to arithmetic properties and often reveals subtle structure.
</INTERNAL_LINK_CANDIDATES> Field axioms (defining rules for addition and multiplication in a field) Binary operation (an operation combining two elements to produce one element) Additive identity (the element 0 under addition) Multiplicative identity (the element 1 under multiplication) Additive inverse (an element that sums with another to give 0) Multiplicative inverse (a nonzero element that multiplies to 1) Distributive law (the rule connecting multiplication and addition) Cancellation law (the property allowing equal terms to be removed from an equation) Rational numbers (the field of fractions of integers) Real numbers (the field of all real-valued quantities) Complex numbers (the field formed by numbers a + bi) Finite field (a field with finitely many elements) Prime field (a finite field of prime order) Field extension (a larger field containing a smaller field) Subfield (a field contained within another field) Characteristic (the number of repeated additions of 1 needed to reach 0, or 0 if none) Algebraically closed field (a field in which every nonconstant polynomial has a root) Field homomorphism (a structure-preserving map between fields) Galois group (the group of field automorphisms fixing a base field) Minimal polynomial (the least-degree monic irreducible polynomial of an algebraic element)