Field (mathematics)

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.

Galois theory, devoted to understanding the symmetries of field extensions, provides an elegant proof of the Abel–Ruffini theorem that general quintic equations cannot be solved in radicals.

One can alternatively define a field by four binary operations (addition, subtraction, multiplication, and division) and their required properties.

For example, the law of distributivity can be proven as follows:[6] The real numbers R, with the usual operations of addition and multiplication, also form a field.

The fields of real and complex numbers are used throughout mathematics, physics, engineering, statistics, and many other scientific disciplines.

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

The requirement 1 ≠ 0 is imposed by convention to exclude the trivial ring, which consists of a single element; this guides any choice of the axioms that define fields.

[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.

[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.

Prior to this, examples of transcendental numbers were known since Joseph Liouville's work in 1844, until Charles Hermite (1873) and Ferdinand von Lindemann (1882) proved the transcendence of e and π, respectively.

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.

For example, It is straightforward to show that, if the ring is an integral domain, the set of the fractions form a field.

[29] The passage from E to E(x) is referred to by adjoining an element to E. More generally, for a subset S ⊂ F, there is a minimal subfield of F containing E and S, denoted by E(S).

The above-mentioned field of rational fractions E(X), where X is an indeterminate, is not an algebraic extension of E since there is no polynomial equation with coefficients in E whose zero is X.

Once again, the field extension E(x) / E discussed above is a key example: if x is not algebraic (i.e., x is not a root of a polynomial with coefficients in E), then E(x) is isomorphic to E(X).

Since every proper subfield of the reals also contains such gaps, R is the unique complete ordered field, up to isomorphism.

The first manifestation of this is at an elementary level: the elements of both fields can be expressed as power series in the uniformizer, with coefficients in Fp.

The primitive element theorem shows that finite separable extensions are necessarily simple, i.e., of the form where f is an irreducible polynomial (as above).

The Lefschetz principle states that C is elementarily equivalent to any algebraically closed field F of characteristic zero.

[55] The theory of modules (the analogue of vector spaces over rings instead of fields) is much more complicated, because the above equation may have several or no solutions.

A widely applied cryptographic routine uses the fact that discrete exponentiation, i.e., computing in a (large) finite field Fq can be performed much more efficiently than the discrete logarithm, which is the inverse operation, i.e., determining the solution n to an equation In elliptic curve cryptography, the multiplication in a finite field is replaced by the operation of adding points on an elliptic curve, i.e., the solutions of an equation of the form Finite fields are also used in coding theory and combinatorics.

The study of function fields and their geometric meaning in higher dimensions is referred to as birational geometry.

The minimal model program attempts to identify the simplest (in a certain precise sense) algebraic varieties with a prescribed function field.

A classical statement, the Kronecker–Weber theorem, describes the maximal abelian Qab extension of Q: it is the field obtained by adjoining all primitive nth roots of unity.

[60] In addition to division rings, there are various other weaker algebraic structures related to fields such as quasifields, near-fields and semifields.

As was mentioned above, commutative rings satisfy all field axioms except for the existence of multiplicative inverses.

Dropping instead commutativity of multiplication leads to the concept of a division ring or skew field;[g] sometimes associativity is weakened as well.

This fact was proved using methods of algebraic topology in 1958 by Michel Kervaire, Raoul Bott, and John Milnor.