# Field

**Type:** Concept  
**Domain:** Algebra  
**Codex URL:** /mathematics/algebra/field/  
**Entry status:** Live — v1.0 (2026-05-27)

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## Curious Level

A field is a set with addition and multiplication where all four arithmetic operations work (including division by any non-zero element). The rational numbers ℚ, real numbers ℝ, and complex numbers ℂ are fields. So are the integers modulo a prime p.

The concept was named "Körper" (body) by Richard Dedekind in 1871, though Galois had worked with finite fields implicitly in 1830.

Fields underlie linear algebra (vector spaces are defined over a field), calculus (ℝ and ℂ), and digital coding (finite fields 𝔽₂, 𝔽₄, etc.).

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## Exploring Level

### Definition

A **field** is a commutative ring (F, +, ·) where every non-zero element has a multiplicative inverse. Equivalently: (F, +) and (F \ {0}, ·) are both abelian groups, and multiplication distributes over addition.

### Examples

| Field | Elements | Characteristic | Notes |
|-------|----------|----------------|-------|
| ℚ | ∞ (countable) | 0 | Smallest char-0 field |
| ℝ | ∞ (uncountable) | 0 | Complete ordered field |
| ℂ | ∞ (uncountable) | 0 | Algebraically closed |
| 𝔽ₚ = ℤ/pℤ | p | p | p must be prime |
| GF(pⁿ) | pⁿ | p | Galois fields; unique up to isomorphism |

### Characteristic

The characteristic of a field is the smallest n with 1+1+…+1 (n times) = 0; or 0 if no such n exists. Every field has characteristic 0 or a prime p.

### Field extensions and Galois theory

A field extension K/F has a Galois group Gal(K/F) of automorphisms of K fixing F. The Fundamental Theorem of Galois Theory gives a bijection between subgroups of Gal(K/F) and intermediate fields, reversing order. This determines which polynomial equations are solvable by radicals.

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## Deep Dive

### Classification of finite fields

For every prime p and n ≥ 1, there is exactly one field of order pⁿ up to isomorphism. No other finite fields exist. The multiplicative group GF(pⁿ)* is cyclic. Wedderburn (1905): every finite division ring is commutative.

### Fundamental Theorem of Galois Theory

For finite Galois extension K/F with G = Gal(K/F): subgroups H ↔ intermediate fields K^H, order-reversing, with |H| = [K : K^H] and [G : H] = [K^H : F]. Normal subgroups correspond to Galois sub-extensions.

### Model theory

The theory of algebraically closed fields of characteristic p is complete and decidable (Tarski, 1949). The theory of ℤ is undecidable (Gödel).

### p-adic fields

ℚₚ is the completion of ℚ under the p-adic absolute value (Hensel, 1897). The collection of all ℚₚ and ℝ are the local fields; global arithmetic is studied via adèles and idèles (Chevalley, 1940).

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## Sources

### Tier 1
- Dummit, D.S. and Foote, R.M. (2004). *Abstract Algebra*. 3rd ed. John Wiley & Sons. [Chapters 13–14]
- Lang, S. (2002). *Algebra*. Revised 3rd ed. Springer. [Chapters V–VI]

### Tier 2
- Stewart, I. (2015). *Galois Theory*. 4th ed. CRC Press.
- Lidl, R. and Niederreiter, H. (1994). *Introduction to Finite Fields and their Applications*. Revised ed. Cambridge University Press.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*