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This article is about an algebraic structure. For vector valued functions, see Vector field. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. 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. 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. Fields can also be defined in different, but equivalent ways. Division by zero is, by definition, excluded. These operations are then subject to the conditions above. Avoiding existential quantifiers is important in constructive mathematics and computing. Rational numbers have been widely used a long time before the elaboration of the concept of field. The abstractly required field axioms reduce to standard properties of rational numbers.

The multiplication of complex numbers can be visualized geometrically by rotations and scalings. It is immediate that this is again an expression of the above type, and so the complex numbers form a field. For example, it was unknown to the Greeks that it is, in general, impossible to trisect a given angle in this way. Not all real numbers are constructible. 2, another problem posed by the ancient Greeks. In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. The field axioms can be verified by using some more field theory, or by direct computation.

In other words, the structure of the binary field is the basic structure that allows computing with bits. This means that every field is an integral domain. Some elementary statements about fields can therefore be obtained by applying general facts of groups. 1 is the identity element of a group that does not contain 0.