the subject of study in field theory, a branch of algebra. The concept of a field is often made use of in branches of mathematics other than algebra.

Four arithmetic operations—the primary operations of addition and multiplication and the inverse operations of subtraction and division—can be performed on ordinary numbers. Generalization yields the concept of a field. Thus, a field is a set of elements for which there are defined two operations—called addition and multiplication—that are subject to the ordinary laws (axioms) of arithmetic:

(I) Addition and multiplication are commutative and associative; that is, a + b = b + a, ab = ba, a + (b + c) = (a + b) + c, and a(bc) = (ab) c.

(II) The set contains an additive identity element 0 (zero); that is, a + 0 = a for any element a in the set. For every a there exists an inverse element —a such that a + (–a) = 0. It thus follows that the subtraction operation a – b can be performed in a field.

(III) The set contains a multiplicative identity element e; that is, ae = a for any a. For each a ≠ 0 there exists an inverse element a^-1 such that aa^-1 = e. It thus follows that division by any a ≠ 0 is possible.

(IV) The operations of addition and multiplication are governed by the distributive law: a(b + c) = ab + ac.

The following are examples of fields.

(1) The set P of all rational numbers.

(2) The set R of all real numbers.

(3) The set K of all complex numbers.

(4) The set of all rational functions in one or several variables with real coefficients.

(5) The set of all numbers of the form , where a and b are rational.

(6) The field of residue classes modulo p, which is defined as follows: Let ρ be a prime number. The integers are divided into classes called residue classes by combining in one class all integers that give the same remainder when divided by p. To define the addition of two classes, take an integer from each of the two classes and form the sum of these integers. The class to which this sum belongs is the sum of the two classes. The product of classes is defined in a similar manner. With these definitions of addition and multiplication the set of classes forms a field containing ρ elements.

It follows from axioms (I) and (II) that the elements of a field form a commutative group with respect to addition and from axioms (I) and (III) that all nonzero elements of a field form a commutative group with respect to multiplication.

If a field contains an element a ≠ 0 such that na = 0 for some integer n, there exists a prime ρ such that pa = 0 for any element a in the field. The field is then said to be of characteristic p; example (6) is an illustration of this case. If na ≠ 0 for any nonzero n and a, the characteristic of the field is zero; the fields in examples (1) through (5) have characteristic zero.

If a subset F of a field G is itself a field with respect to the addition and multiplication in G, then F is called a subfield of G and G an extension of F. A field that does not have subfields is called a prime field. All prime fields are exhausted by the fields of examples (1) and (6) (for all possible choices of the prime p). Every field contains a unique prime subfield; the prime field in each of the fields in examples (2) through (5) is the field of rational numbers. The following is a “natural” problem: Describe all extensions of a given prime field. Steinitz’ theorem, which is given below, deals with just this problem.

Two types of extensions have a comparatively simple structure. One type is the simple transcendental extension, where G is the field of all rational functions of one variable with coefficients in P. The other type is the simple algebraic extension, an illustration of which is the field of example (5). Such an extension is obtained by a construction similar to that of example (6): if f(x) is a polynomial of degree n that is irreducible in F, G is the field of residue classes of polynomials of degree n modulo f(x). To obtain a simple algebraic extension of the second type, we in effect adjoin to F a root of f(x) and all elements that can be expressed in terms of this root and the elements of F. If every element of G is a root of some polynomial with coefficients in F, G is said to be an algebraic extension of F. According to Steinitz’ theorem, any extension of a field F can be obtained in two stages: first we obtain a transcendental extension T of F (by forming the field of rational functions, not necessarily of one variable, over F) and then an algebraic extension of T. Only fields in which every polynomial decomposes into a product of linear factors do not have algebraic extensions. Such fields are called algebraically closed fields. The field of complex numbers is algebraically closed (the fundamental theorem of algebra). Every field is a subfield of an algebraically closed field.

Certain types of fields have been studied in particular detail. The theory of algebraic numbers deals primarily with simple algebraic extensions of the field of rational numbers. The theory of algebraic functions investigates simple algebraic extensions of the field of rational functions with complex coefficients; considerable attention is paid to finite extensions of the field of rational functions over an arbitrary field of constants, that is, rational functions with coefficients in some arbitrary given field. Finite extensions of fields, particularly their automorphisms, are studied in Galois theory; many problems that arise in the solution of algebraic equations find an answer here. In many problems of algebra, especially in various branches of the theory of fields, normed fields play an important role. Ordered fields appear and have been studied in connection with geometric investigations.

in biology, a concept that describes a biological system the behavior of whose parts is determined by the position of the parts in the system. The existence of such systems has been determined principally by numerous experiments on the movement, elimination, and addition of parts in the embryo. In many cases, normal organisms develop from such embryos, since their components change their path of development according to their new position in the whole.

Between 1912 and 1922, A. G. Gurvich introduced the concept of field (morphogenetic field) into embryology and established the task of elucidating its laws. He identified these laws initially with an indivisible factor governing morphogenesis and later with a system of intercellular interactions that determine the movement and differentiation of embryonic cells. In 1925 the Austrian scientist P. Weiss applied the concept of field to the processes of regeneration, and in 1934 the British scientists J. Huxley and G. de Beer combined the concept with the concept of gradient. In the 1940’s through 1960’s, the British biologist C. Waddington and the French mathematician R. Thorn introduced concepts of embryonic development as a vector field divided into a limited number of zones of “structural stability.”

The range of concepts of field is being intensively developed in contemporary theoretical biology, but no unified opinion as to the intrinsic principles underlying the phenomena described by the concept of field has been worked out.

in physics, a special form of matter. A field is a physical system that has an infinite number of degrees of freedom. Examples are electromagnetic fields, gravitational fields, the field of nuclear forces, and the quantized fields associated with different particles.

The concept of electric and magnetic fields was introduced in the 1830’s by M. Faraday, who adopted the concept as an alternative to the theory of long-range interaction, that is, the interaction of particles at a distance without any intermediate agent. The electrostatic interaction of charged particles according to Coulomb’s law and the gravitational interaction of bodies according to Newton’s law of universal gravitation, for example, had been interpreted in terms of action at a distance. The field concept was a resurrection of the theory of local interaction that had been proposed by R. Descartes in the first half of the 17th century. In the 1860’s, J. C. Maxwell developed Faraday’s idea of the electromagnetic field and mathematically formulated its laws.

According to the field concept, the particles participating in an interaction, such as an electromagnetic or gravitational interaction, create at each point in the space surrounding them a special state—a force field manifested in the action of a force on other particles placed at a point in this space. The first interpretation of a field to be proposed was a mechanistic one: the field was viewed as the elastic stresses of a hypothetical medium called the ether. An ether with the properties of an elastic medium, however, proved to be in sharp contradiction with the results of subsequent experiments. From the modern point of view, such a mechanistic interpretation of fields is in general meaningless, since the elastic properties of macroscopic bodies are themselves explained entirely by the electromagnetic interactions of the particles making up these bodies. The theory of relativity rejected the concept of the ether as a special elastic medium and ascribed fundamental importance to fields as a primary physical reality. Indeed, according to the theory of relativity, the rate of propagation of any interaction cannot exceed the speed of light in a vacuum. In a system of interacting particles, therefore, a force acting at a given moment on a particle in the system is not determined by the position of other particles at the same moment; that is, a change in the position of one particle affects another particle not immediately but only after some time interval. Thus, the interaction of particles whose relative speed is comparable to the speed of light can be described only in terms of the fields generated by the particles. A change in the state or position of a particle results in a change in the field generated by it. This change is reflected in another particle only after the finite time interval necessary for the change to propagate to the second particle.

Not only do fields realize interactions between particles, but free fields can exist and appear independently of the particles that generated them. This is true, for example, of electromagnetic waves. It therefore is clear that fields should be considered as a special form of matter.

To each type of interaction in nature there correspond certain fields. The description of fields in classical (nonquantum) field theory is accomplished by means of one or more continuous field functions that depend on the position coordinates of the point (x, y, z) at which the field is considered and on the time t. Thus, an electromagnetic field can be completely described by using four functions: the scalar potential ɸ(x, y, z, t) and the vector potential A(x, y, z, t), which together form a single four-dimensional vector in space-time. The strengths of electric and magnetic fields are expressed in terms of the derivatives of these functions. In the general case the number of independent field functions is determined by the number of internal degrees of freedom of the particles corresponding to the given field (see below)—for example, the particles’ spin and isotopic spin. On the basis of general principles—the requirements of relativistic invariance and of some more specific assumptions, such as the superposition principle and gauge invariance for an electromagnetic field—an expression for action can be formed from the field functions, and the principle of least action can be used to obtain differential equations defining the field. The values of the field functions at each individual point can be regarded as the generalized coordinates of the field. Consequently, the field is a physical system with an infinite number of degrees of freedom. In accordance with the general laws of mechanics, it is possible to obtain an expression for the generalized momenta of the field and to determine the densities of its energy, momentum, and angular momentum.

Experiment has shown—initially for the electromagnetic field —that the energy and momentum of a field vary discretely. In other words, fields can be associated with specific particles—for example, an electromagnetic field with photons, and a gravitational field with gravitons. The description of fields by means of field functions is thus only an approximation with a certain range of applicability. In order to take into account the discrete properties of fields—that is, to construct a quantum field theory—the generalized coordinates and momenta of fields must be regarded not as numbers but as operators for which certain commutation relations are satisfied. It may be noted that the transition from classical to quantum mechanics is made in a similar manner.

Quantum mechanics shows that a system of interacting particles can be described by means of some quantum field. Thus, not only are certain particles associated with every field, but, conversely, all known particles are associated with quantized fields. This fact is an example of the wave-particle duality of matter. Quantized fields describe the annihilation (or creation) of particles and the simultaneous production (or annihilation) of antiparticles. The electron-positron field in quantum electrodynamics is an example of such a field.

The type of the commutation relations for field operators depends on the kind of particles that correspond to the given field. W. Pauli showed in 1940 that field operators commute for particles with integral spin, and these particles obey Bose-Ein-stein statistics. For particles with half-integral spin, the operators anticommute, and the corresponding particles obey Fermi-Dirac statistics. If the particles obey Bose-Einstein statistics—for example, photons and gravitons—then many particles can occupy the same quantum state; in the limiting case infinitely many particles can be in the same quantum state. At this limit the mean values of the quantized fields become ordinary classical fields—for example, classical electromagnetic and gravitational fields describable by continuous functions of the position coordinates and time. Corresponding classical fields do not exist for fields associated with particles having half-integral spin.

The present-day theory of elementary particles is constructed as a theory of interacting quantum fields, such as electron-positron, photon, and meson fields.