Field (mathematics) - Generalizations

Generalizations

There are also proper classes with field structure, which are sometimes called Fields, with a capital F:

• The surreal numbers form a Field containing the reals, and would be a field except for the fact that they are a proper class, not a set.
• The nimbers form a Field. The set of nimbers with birthday smaller than 22n, the nimbers with birthday smaller than any infinite cardinal are all examples of fields.

In a different direction, differential fields are fields equipped with a derivation. For example, the field R(X), together with the standard derivative of polynomials forms a differential field. These fields are central to differential Galois theory. Exponential fields, meanwhile, are fields equipped with an exponential function that provides a homomorphism between the additive and multiplicative groups within the field. The usual exponential function makes the real and complex numbers exponential fields, denoted R_exp and C_exp respectively.

Generalizing in a more categorical direction yields the field with one element and related objects.