Trait Field 

pub trait Field:
    CommutativeRing
    + InvMonoid
    + Div<Output = Self>
    + DivAssign { }

Represents a Field in abstract algebra.

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 do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

Mathematical Definition

A field is a CommutativeRing where every non-zero element has a multiplicative inverse. This means it satisfies the following laws:

1. CommutativeRing Laws:

   • Forms an AbelianGroup under addition (associative, commutative, identity 0, inverses).
   • Forms a MulMonoid under multiplication (associative, identity 1).
   • Multiplication is commutative (a * b = b * a).
   • Multiplication distributes over addition (a * (b + c) = a*b + a*c).

2. Multiplicative Inverse:

   • For every element a not equal to 0, there exists an element a⁻¹ such that a * a⁻¹ = 1. (Provided by the InvMonoid trait)

Examples

• Real numbers (f32, f64)
• Complex numbers (Complex<T>)
• Rational numbers (not implemented in this crate)

Counter-examples

• Integers (i32, i64): Lack multiplicative inverses for most elements.
• Quaternions (Quaternion<T>): Multiplication is not commutative.

Dyn Compatibility

This trait is not dyn compatible.

In older versions of Rust, dyn compatibility was called "object safety".

Implementors

impl<T> Field for T

where T: CommutativeRing + InvMonoid + Div<Output = Self> + DivAssign,