A field is a set equipped with addition and multiplication satisfying rules that make subtraction and division by nonzero elements possible. Familiar examples include the rational, real, and complex numbers, but fields can also contain finitely many elements. In abstract algebra, the concept identifies the algebraic properties shared by these systems without requiring their elements to be ordinary numbers. (math.mit.edu)
Definition and axioms
A field (F) has two binary operations, addition (+) and multiplication (\cdot), and distinct elements (0) and (1). Its defining axioms require:
- Associativity: ((a+b)+c=a+(b+c)) and ((ab)c=a(bc)).
- Commutativity: (a+b=b+a) and (ab=ba).
- Identity elements: (a+0=a) and (a\cdot1=a).
- Additive inverses: every (a) has an element (-a) satisfying (a+(-a)=0).
- Multiplicative inverses: every (a\ne0) has an element (a^{-1}) satisfying (aa^{-1}=1).
- Distributivity: (a(b+c)=ab+ac).
Both operations take elements of (F) to elements of (F). Subtraction means adding an additive inverse; division means multiplying by a multiplicative inverse. Division by zero is not defined. (math.mit.edu)
Equivalently, a field is a commutative ring with (1\ne0) in which every nonzero element is invertible. Its additive structure and its nonzero multiplicative structure are abelian groups. Dropping commutativity of multiplication gives the broader concept of a division ring. (math.mit.edu)
Examples and non-examples
The rational numbers (\mathbb Q), real numbers (\mathbb R), and complex numbers (\mathbb C) are fields with their usual operations. The integers (\mathbb Z) are not: for example, (2) has no multiplicative inverse in (\mathbb Z). Likewise, square real matrices of size at least two do not form a field, because some nonzero matrices are not invertible. (homepages.ucl.ac.uk)
For a prime number (p), the residue classes (\mathbb Z/p\mathbb Z) form a field, denoted (\mathbb F_p), using modular arithmetic. In (\mathbb F_5), for example, (3+4=2), and (2^{-1}=3) because (2\cdot3=1) modulo (5). If the modulus is composite, the resulting ring is not a field: modulo (6), the nonzero classes (2) and (3) have product zero. (homepages.ucl.ac.uk)
Characteristic and finite fields
The characteristic of a field is the least positive integer (n) for which adding (1) to itself (n) times gives zero. If no such integer exists, the characteristic is zero. A positive characteristic must be prime. Every field contains a smallest subfield, its prime subfield, isomorphic to (\mathbb Q) in characteristic zero or (\mathbb F_p) in characteristic (p). (jmilne.org)
Every finite field has (p^n) elements for some prime (p) and positive integer (n). Conversely, for every prime power (p^n), a field of that size exists and is unique up to isomorphism—a bijection preserving the field operations. It is written (\mathbb F_{p^n}) or (\operatorname{GF}(p^n)). Its characteristic is (p), not (p^n). (math.mit.edu)
Such fields can be constructed by taking polynomials over (\mathbb F_p) modulo an irreducible polynomial of degree (n). For example, [ \mathbb F_4\cong\mathbb F_2[x]/(x^2+x+1). ] Writing (\alpha) for the class of (x), its elements are (0,1,\alpha,1+\alpha), with (\alpha^2=\alpha+1). This is not the ring of integers modulo four. (math.mit.edu)
Subfields and extensions
A subfield is a subset that is itself a field under the inherited operations. When (K) is a subfield of (L), (L/K) is a field extension. The larger field is a vector space over the smaller one; its dimension, denoted ([L:K]), is the extension’s degree. (jmilne.org)
An element of an extension is algebraic over (K) if it satisfies a nonzero polynomial with coefficients in (K); otherwise it is transcendental. For instance, [ \mathbb Q(\sqrt2)={a+b\sqrt2:a,b\in\mathbb Q} ] has degree two over (\mathbb Q). A field is algebraically closed if every nonconstant polynomial over it has a root in it. The fundamental theorem of algebra establishes this property for (\mathbb C). (jmilne.org)
Linear algebra and applications
Fields provide the scalars for linear algebra. Vectors, matrices, and linear equations can be studied over any field, not only over real or complex numbers. The availability of inverses for nonzero scalars makes division-based manipulations possible; choosing a different field changes the arithmetic without abandoning the underlying framework. (homepages.ucl.ac.uk)
Galois theory studies field extensions through their operation-preserving symmetries. For a finite Galois extension, its fundamental theorem relates intermediate fields to subgroups of the associated Galois group, connecting field structure with group theory. (jmilne.org)
A concrete application occurs in cryptography: the Advanced Encryption Standard interprets bytes as elements of (\mathbb F_{256}). Addition is bitwise exclusive OR, while multiplication uses binary-coefficient polynomials reduced modulo a specified irreducible polynomial. These are field operations, not ordinary integer arithmetic modulo (256). (nvlpubs.nist.gov)