AlgebraicField

A field is a set on which addition, subtraction, multiplication, and division are defined, and behave basically like those operations on the real numbers. More precisely, a field is a commutative group under its addition, the non-zero elements of the field form a commutative group under its multiplication, and the distributitve law holds.

Some common examples of fields include:

• the rational numbers

• the real numbers

• the complex numbers

• the integers modulo a prime

The most familiar example of a thing that is not a field is the integers. This may be surprising, since integers seem to have addition, subtraction, multiplication and division. Why don’t they form a field?

Because integer multiplication does not form a group; it’s commutative and associative, but integers do not have multiplicative inverses. I.e. if a is any integer other than 1 or -1, there is no integer b such that a*b = 1. The existence of inverses is requried to form a field.

If a type T conforms to the Real protocol, then T and Complex<T> both conform to AlgebraicField.

See Also:

• Real

• SignedNumeric

• Numeric

• AdditiveArithmetic