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", so this trait is not object safe.

Implementors

impl<T> Field for T

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