A field is a set with two operations — addition and multiplication — where both non-zero elements under multiplication and all elements under addition form abelian groups, and multiplication distributes over addition. Fields are the number systems where all four arithmetic operations work: ℚ, ℝ, ℂ, and finite fields 𝔽ₚ are the canonical examples.
What is a field?
The rational numbers ℚ = {…, −1, 0, ½, 1, 2, …} let you add, subtract, multiply, and divide (by anything except zero). You can undo both addition and multiplication. This is special — the integers ℤ don't allow this (you can't divide 1 by 2 inside ℤ).
A field is the abstract structure where all four arithmetic operations are well-behaved: add, subtract, multiply, divide (by non-zero). The rational numbers ℚ, the real numbers ℝ, and the complex numbers ℂ are all fields. So are the integers modulo a prime p — for example, {0, 1, 2, 3, 4} with addition and multiplication mod 5.
Fields are important because:
Linear algebra lives in fields — vectors, matrices, and solving equations all work over any field
Calculus lives in ℝ or ℂ — but the abstract structure is that of a field
Coding theory uses finite fields — digital communications, CDs, QR codes all use the field 𝔽₂ = {0, 1}
Galois theory connects fields to groups — which polynomial equations are solvable depends on the structure of field extensions
The concept of a field was isolated and named ("Körper" — body) by Richard Dedekind in 1871, though Galois had worked with finite fields implicitly in 1830.
Formal definition and structure
Definition
A field is a commutative ring (F, +, ·) in which every non-zero element has a multiplicative inverse: for all a ∈ F with a ≠ 0, there exists a⁻¹ ∈ F such that a · a⁻¹ = 1.
Equivalently: (F, +) is an abelian group with identity 0; (F \ {0}, ·) is an abelian group with identity 1; multiplication distributes over addition.
Examples
Field
Elements
Characteristic
Notes
ℚ
∞ (countable)
0
Rationals; smallest field of char 0
ℝ
∞ (uncountable)
0
Reals; Dedekind-complete ordered field
ℂ
∞ (uncountable)
0
Algebraically closed
𝔽ₚ = ℤ/pℤ
p (finite)
p
p must be prime
𝔽_{pⁿ} (GF(pⁿ))
pⁿ (finite)
p
Galois fields; unique up to isomorphism
Characteristic
The characteristic of a field F is the smallest positive integer n such that 1 + 1 + … + 1 (n times) = 0; if no such n exists, the characteristic is 0. Every field has characteristic 0 or a prime p. Fields of characteristic 0 contain ℚ as a subfield; fields of characteristic p contain 𝔽ₚ as a subfield.
Field extensions
A field extension K/F is a pair of fields with F ⊆ K. The extension ℂ/ℝ has degree 2 (ℂ = ℝ(i) with minimal polynomial x²+1 over ℝ). The extension ℝ/ℚ has infinite degree. Field extensions are the arena of Galois theory: the Galois group Gal(K/F) is the group of field automorphisms of K fixing F pointwise.
Algebraic closure
A field F is algebraically closed if every non-constant polynomial in F[x] has a root in F. ℂ is algebraically closed (Fundamental Theorem of Algebra); ℝ is not (x²+1 has no real root). Every field has an algebraic closure, unique up to isomorphism.
Classification of fields, finite fields, and Galois theory
Classification of finite fields
Theorem (Galois / Steinitz): For every prime p and every positive integer n, there exists exactly one field (up to isomorphism) with pⁿ elements, denoted 𝔽_{pⁿ} or GF(pⁿ). There are no other finite fields. The multiplicative group 𝔽_{pⁿ}* is cyclic of order pⁿ − 1.
Finite fields were first constructed and studied by Galois (1830). Wedderburn's theorem (1905) shows that every finite division ring is commutative — every finite skew field is a field.
Fundamental Theorem of Galois Theory
For a finite Galois extension K/F with Galois group G = Gal(K/F):
There is a one-to-one, order-reversing correspondence between subgroups of G and intermediate fields F ⊆ E ⊆ K
H ↔ K^H (fixed field of H); E ↔ Gal(K/E)
Normal subgroups H ◁ G correspond to Galois extensions E/F
[K : E] = |H| and [E : F] = [G : H]
This establishes the Galois correspondence — the bridge between field theory and group theory that resolves which polynomial equations are solvable.
Model theory and field axioms
The theory of algebraically closed fields of characteristic p is complete and decidable (Tarski, 1949) — every first-order sentence in the language of rings is either provable or disprovable from the axioms. This is the model-theoretic underpinning of Chevalley's theorem and Grothendieck's approach to algebraic geometry. By contrast, the theory of ℤ is undecidable (Gödel's incompleteness).
p-adic fields
Beyond ℚ and its completions ℝ, there is a field ℚₚ for each prime p — the p-adic numbers (Hensel, 1897). ℚₚ is the completion of ℚ with respect to the p-adic absolute value |a/b|ₚ = p^{v_p(b)−v_p(a)}, where v_p denotes p-adic valuation. The p-adic fields are the local fields over ℚ, and the study of arithmetic over all primes simultaneously — "global" arithmetic — is via adèles and idèles (Chevalley, 1940).
Sources
Dummit, D.S. and Foote, R.M. (2004). Abstract Algebra. 3rd ed. John Wiley & Sons. — Chapters 13–14 cover field theory and Galois theory. Tier 1
Lang, S. (2002). Algebra. Revised 3rd ed. Springer. — Chapters V–VI on fields and Galois theory. Tier 1
Stewart, I. (2015). Galois Theory. 4th ed. CRC Press. — Accessible treatment of field extensions and Galois correspondence. Tier 2
Lidl, R. and Niederreiter, H. (1994). Introduction to Finite Fields and their Applications. Revised ed. Cambridge University Press. — Standard reference for finite fields. Tier 2
Relationships
SpecialisesRing — a field is a ring where every non-zero element is a unit
SpecialisesGroup — both (F, +) and (F*, ·) are abelian groups
Key applicationGalois Theory — Galois group of a field extension determines polynomial solvability
Key application Linear algebra — vector spaces are defined over a field
Key application Coding theory — error-correcting codes use finite fields
Key person Richard Dedekind (1831–1916) — named and defined "Körper" (field), 1871
Key person Évariste Galois (1811–1832) — finite fields and Galois theory