In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.
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The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements.
The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that angle trisection and squaring the circle cannot be done with a compass and straightedge. Moreover, it shows that quintic equations are, in general, algebraically unsolvable.
Fields serve as foundational notions in several mathematical domains. This includes different branches of mathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Number fields, the siblings of the field of rational numbers, are studied in depth in number theory. Function fields can help describe properties of geometric objects.
An equivalent, and more succinct, definition is: a field has two commutative operations, called addition and multiplication; it is a group under addition with 0 as the additive identity; the nonzero elements are a group under multiplication with 1 as the multiplicative identity; and multiplication distributes over addition.
In antiquity, several geometric problems concerned the (in)feasibility of constructing 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 in this way. These problems can be settled using the field of constructible numbers.[7] Real constructible numbers are, by definition, lengths of line segments that can be constructed from the points 0 and 1 in finitely many steps using only compass and straightedge. These numbers, endowed with the field operations of real numbers, restricted to the constructible numbers, form a field, which properly includes the field Q of rational numbers. The illustration shows the construction of square roots of constructible numbers, not necessarily contained within Q. Using the labeling in the illustration, construct the segments AB, BD, and a semicircle over AD (center at the midpoint C), which intersects the perpendicular line through B in a point F, at a distance of exactly h = p \displaystyle h=\sqrt p from B when BD has length one.
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 above), and I is the multiplicative identity (denoted 1 in the axioms above). The field axioms can be verified by using some more field theory, or by direct computation. For example,
This field is called a finite field or Galois field with four elements, and is denoted F4 or GF(4).[8] The subset consisting of O and I (highlighted in red in the tables at the right) is also a field, known as the binary field F2 or GF(2). In the context of computer science and Boolean algebra, O and I are often denoted respectively by false and true, and the addition is then denoted XOR (exclusive or). In other words, the structure of the binary field is the basic structure that allows computing with bits.
The axioms of a field F imply that it is an abelian group under addition. This group is called the additive group of the field, and is sometimes denoted by (F, +) when denoting it simply as F could be confusing.
then F is said to have characteristic 0.[11] For example, the field of rational numbers Q has characteristic 0 since no positive integer n is zero. Otherwise, if there is a positive integer n satisfying this equation, the smallest such positive integer can be shown to be a prime number. It is usually denoted by p and the field is said to have characteristic p then.For example, the field F4 has characteristic 2 since (in the notation of the above addition table) I + I = O.
is compatible with the addition in F (and also with the multiplication), and is therefore a field homomorphism.[12] The existence of this homomorphism makes fields in characteristic p quite different from fields of characteristic 0.
A field is called a prime field if it has no proper (i.e., strictly smaller) subfields. Any field F contains a prime field. If the characteristic of F is p (a prime number), the prime field is isomorphic to the finite field Fp introduced below. Otherwise the prime field is isomorphic to Q.[14]
Finite fields (also called Galois fields) are fields with finitely many elements, whose number is also referred to as the order of the field. The above introductory example F4 is a field with four elements. Its subfield F2 is the smallest field, because by definition a field has at least two distinct elements, 0 and 1.
Such a splitting field is an extension of Fp in which the polynomial f has q zeros. This means f has as many zeros as possible since the degree of f is q. For q = 22 = 4, it can be checked case by case using the above multiplication table that all four elements of F4 satisfy the equation x4 = x, so they are zeros of f. By contrast, in F2, f has only two zeros (namely 0 and 1), so f does not split into linear factors in this smaller field. Elaborating further on basic field-theoretic notions, it can be shown that two finite fields with the same order are isomorphic.[16] It is thus customary to speak of the finite field with q elements, denoted by Fq or GF(q).
Historically, three algebraic disciplines led to the concept of a field: the question of solving polynomial equations, algebraic number theory, and algebraic geometry.[17] A first step towards the notion of a field was made in 1770 by Joseph-Louis Lagrange, who observed that permuting the zeros x1, x2, x3 of a cubic polynomial in the expression
(with ω being a third root of unity) only yields two values. This way, Lagrange conceptually explained the classical solution method of Scipione del Ferro and François Viète, which proceeds by reducing a cubic equation for an unknown x to a quadratic equation for x3.[18] Together with a similar observation for equations of degree 4, Lagrange thus linked what eventually became the concept of fields and the concept of groups.[19] Vandermonde, also in 1770, and to a fuller extent, Carl Friedrich Gauss, in his Disquisitiones Arithmeticae (1801), studied the equation
for a prime p and, again using modern language, the resulting cyclic Galois group. Gauss deduced that a regular p-gon can be constructed if p = 22k + 1. Building on Lagrange's work, Paolo Ruffini claimed (1799) that quintic equations (polynomial equations of degree 5) cannot be solved algebraically; however, his arguments were flawed. These gaps were filled by Niels Henrik Abel in 1824.[20] Évariste Galois, in 1832, devised necessary and sufficient criteria for a polynomial equation to be algebraically solvable, thus establishing in effect what is known as Galois theory today. Both Abel and Galois worked with what is today called an algebraic number field, but conceived neither an explicit notion of a field, nor of a group.
In 1871 Richard Dedekind introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, the German word Körper, which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by Moore (1893).[21]
By a field we will mean every infinite system of real or complex numbers so closed in itself and perfect that addition, subtraction, multiplication, and division of any two of these numbers again yields a number of the system.
The first clear definition of an abstract field is due to Weber (1893).[24] In particular, Heinrich Martin Weber's notion included the field Fp. Giuseppe Veronese (1891) studied the field of formal power series, which led Hensel (1904) to introduce the field of p-adic numbers. Steinitz (1910) synthesized the knowledge of abstract field theory accumulated so far. He axiomatically studied the properties of fields and defined many important field-theoretic concepts. The majority of the theorems mentioned in the sections Galois theory, Constructing fields and Elementary notions can be found in Steinitz's work. Artin & Schreier (1927) linked the notion of orderings in a field, and thus the area of analysis, to purely algebraic properties.[25] Emil Artin redeveloped Galois theory from 1928 through 1942, eliminating the dependency on the primitive element theorem.
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