In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed in a way that satisfies some familiar rules from the arithmetic of ordinary numbers.

All fields are rings, but not conversely. Fields differ from rings most importantly in the requirement that division be possible, but also, in modern definitions, by the requirement that the multiplication operation in a field be commutative. A ring in which division is possible but isn't assumed to be commutative is called a division ring (sometimes also called a skew field), although historically division rings were called fields and fields were commutative fields.

The prototypical example of a field is Q, the field of rational numbers. Other important examples include the field of real numbers R, the field of complex numbers C and, for any power q = pⁿ of a prime number p, the (unique) finite field with q elements, denoted F_q or GF(q). For any field K, the set K(X) of rational functions with coefficients in K is also a field.

The mathematical discipline concerned with the study of fields is called field theory.

A field is a specific type of integral domain, and can be characterized by the following (not necessarily exhaustive) chain of class inclusions:

• Commutative rings ⊃ integral domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields

Definition and illustration

An example of a field is the set Q of rational numbers. In Q, there are four essential operations: addition together with subtraction, and multiplication with division. Intuitively, a field is a set of numbers which has four such operations. In order to qualify as a field, these operations have to satisfy certain axioms.

A field is a set together with two operations, usually called addition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold:

Closure of F under addition and multiplication

For all a, b in F, both a + b and a · b are in F (or more formally, + and · are binary operations on F).

Associativity of addition and multiplication

For all a, b, and c in F, the following equalities hold: a + (b + c) = (a + b) + c and a · (b · c) = (a · b) · c.

Commutativity of addition and multiplication

For all a and b in F, the following equalities hold: a + b = b + a and a · b = b · a.

Additive and multiplicative identity

There exists an element of F, called the additive identity element and denoted by 0, such that for all a in F, a + 0 = a. Likewise, there is an element, called the multiplicative identity element and denoted by 1, such that for all a in F, a · 1 = a. For technical reasons, the additive identity and the multiplicative identity are required to be distinct.

Additive and multiplicative inverses

For every a in F, there exists an element −a in F, such that a + (−a) = 0. Similarly, for any a in F other than 0, there exists an element a⁻¹ in F, such that a · a⁻¹ = 1. (The elements a + (−b) and a · b⁻¹ are also denoted a − b and a/b, respectively.) In other words, subtraction and division operations exist.

Distributivity of multiplication over addition

For all a, b and c in F, the following equality holds: a · (b + c) = (a · b) + (a · c).

First example: rational numbers

The easiest example for a field are the rational numbers consisting of fractions a/b, where a and b are integers, and b ≠ 0. The additive inverse of such a fraction is simply −a/b, and the multiplicative inverse—provided that a ≠ 0, as well—is b/a. To see the latter note that

The abstractly required field axioms reduce to standard properties of rational numbers, such as the law of distributivity

or the law of commutativity and law of associativity.

Second example: a field with four elements

+ O I A B
O O I A B
I I O B A
A A B O I
B B A I O
· O I A B
O O O O O
I O I A B
A O A B I
B O B I A

In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. The following example is a field consisting of four elements called O, I, A and B. The notation is chosen such that O plays the role of the additive identity element (denoted 0 in the axioms), and I is the multiplicative identity (denoted 1 above). Checking that all field axioms are indeed satisfied is easy, if tedious. For example:

A · (B + A) = A · 1 = A, which equals A · B + A · A = I + B = A, as required by the distributivity.

The above field is called a finite field with four elements, denoted F₄. Field theory is concerned with understanding the reasons for the existence of this field, defined in a fairly ad-hoc manner, and with describing its inner structure. For example, from a glance at the multiplication table, it can be seen that any non-zero element, i.e. I, A, and B is a power of A. Indeed A = A¹, B = A² = A · A, and finally I = A³ = A · A · A. This is no coincidence, but one of the starting points of a deeper understanding of (finite) fields.

Related algebraic structures

Ring and field axioms
Abelian group Ring Commutative
ring Skew field or
Division ring Field
Abelian (additive) group
structure Yes Yes Yes Yes Yes
Multiplicative structure
and distributivity – Yes Yes Yes Yes
Commutativity of multiplication – No Yes No Yes
Multiplicative inverses – No No Yes Yes

The axioms imposed above resemble the ones familiar from other algebraic structures. For example, the existence of the binary operation "·", together with its commutativity, associativity, (multiplicative) identity element and inverses are precisely the axioms for an abelian group. In other words, for any field, the subset of nonzero elements F \ {0}, also often denoted F^×, is an abelian group (F^×, ·) usually called multiplicative group of the field. Likewise (F, +) is an abelian group. The structure of a field is hence the same as specifying such two group structures (on the same set), obeying the distributivity.

Important other algebraic structures such as rings arise when requiring only part of the above axioms. For example, if the requirement of commutativity of the multiplication operation · is dropped, one gets structures usually called division rings or skew fields.

Remarks

By elementary group theory, applied to the abelian groups (F^×, ·), and (F, +), the additive inverse −a and the multiplicative inverse a⁻¹ are uniquely determined by a.

Similar direct consequences from the field axioms include

−(a · b) = (−a) · b = a · (−b), in particular −a = (−1) · a

as well as

a · 0 = 0.

Both can be shown by replacing b or c with 0 in the distributive property

History

The concept of field was used implicitly by Niels Henrik Abel and Évariste Galois in their work on the solvability of polynomial equations with rational coefficients of degree 5 or higher.

In 1871, Richard Dedekind called a set of real or complex numbers which is closed under the four arithmetic operations a "field". He used the German word Körper – "body" for this notion, hence the common use of the letter K to denote a field. He also defined rings (then called order or order-modul), but the term "a ring" (Zahlring) was invented by Hilbert. [1]

In 1881, Leopold Kronecker defined what he called a "domain of rationality", which is indeed a field of polynomials in modern terms. In 1893, Heinrich M. Weber gave the first clear definition of an abstract field.[2] In 1910 Ernst Steinitz published the very influential paper Algebraische Theorie der Körper (English: Algebraic Theory of Fields).[3] In this paper he axiomatically studies the properties of fields and defines many important field theoretic concepts like prime field, perfect field and the transcendence degree of a field extension.

Emil Artin developed the relationship between groups and fields in great detail during 1928-1942.

Examples

Rationals and algebraic numbers

The field of rational numbers Q has been introduced above. A related class of fields very important in number theory are algebraic number fields. We will first give an example, namely the field Q[ζ₃] consisting of expressions

a + b · ζ + c · ζ², a, b, c ∈ Q

where ζ is a third root of unity, i.e. a complex number satisfying ζ³ = 1, ζ ≠ 1, can be used to prove a special case of Fermat's last theorem, which asserts the non-existence of rational nonzero solutions to the equation

x³ + y³ = z³.

In the language of field extensions detailed below, Q[ζ₃] is a field extension of degree 3. Algebraic number fields are by definition finite field extensions of Q, that is, fields containing Q having finite dimension as a Q-vector space.

Reals, complex numbers, and p-adic numbers

Take the real numbers R, under the usual operations of addition and multiplication. When the real numbers are given the usual ordering, they form a complete ordered field; it is this structure which provides the foundation for most formal treatments of calculus.

The complex numbers C consist of expressions

a + bi

where i is the imaginary unit, i.e. a (non-real) number satisfying i² = −1. Addition and multiplication of real numbers are defined in such a way that all field axioms hold for C. For example, the distributive law enforces

(a + bi)·(c + di) = ac + bci + adi + bdi², which equals ac−bd + (bc + ad)i.

The real numbers can be constructed by completing the rational numbers, i.e. filling the "gaps": for example √2 is such a gap. By a formally very similar procedure, another important class of fields, the field of p-adic numbers Q_p is built. It is used in number theory and p-adic analysis.

Hyperreal numbers and superreal numbers extend the real numbers with the addition of infinitesimal and infinite numbers.

Constructible numbers

In antiquity, several geometric problems concerned the (in)feasibility to construct certain numbers with compass and straightedge. For example it was unknown to the Greeks that it is in general impossible to trisect a given angle. Using the field notion and field theory allows to settle these problems. To do so, the field of constructible numbers is considered. It contains, on the plane, the points 0 and 1, and all complex numbers that can be constructed from these two by a finite number of construction steps using only compass and straightedge. This set, endowed with the usual addition and multiplication of complex numbers does form a field. For example, multiplying two (real) numbers r₁ and r₂ that have already been constructed can be done using construction at the right, based on the intercept theorem. This way, the obtained field F contains all rational numbers, but is bigger than Q, because for any f ∈ F, the square root of f is also a constructible number.

Finite fields

Finite fields (also called Galois fields) are fields with finitely many elements. The above introductory example F₄ is a field with four elements. Highlighted in the multiplication and addition tables above is the field F₂ consisting of two elements O and I. This is the smallest field, because by definition a field has at least two distinct elements 1 ≠ 0. Interpreting the addition and multiplication in this latter field as XOR and Logical AND operation, this field finds applications in computer science, especially in cryptography and coding theory.