Module scirust::algebra::structure::field

Defines the field algebraic structure.

A field is a set together with two operations, usually called addition and multiplication, and denoted by + and *, respectively, such that the following axioms hold; subtraction and division are defined in terms of the inverse operations of addition and multiplication:

Closure of F under addition and multiplication

• For all a, b in F, both a + b and a * b are in F (or more formally, + and * are binary operations on F).

Associativity of addition and multiplication

• For all a, b, and c in F, the following equalities hold: a + (b + c) = (a + b) + c and a * (b * c) = (a * b) * c.

Commutativity of addition and multiplication

• For all a and b in F, the following equalities hold: a + b = b + a and a * b = b * a.

Existence of additive and multiplicative identity elements

• There exists an element of F, called the additive identity element and denoted by 0, such that for all a in F, a + 0 = a.
• Likewise, there is an element, called the multiplicative identity element and denoted by 1, such that for all a in F, a * 1 = a.
• To exclude the trivial ring, the additive identity and the multiplicative identity are required to be distinct.

Existence of additive inverses and multiplicative inverses

• For every a in F, there exists an element -a in F, such that a + (-a) = 0.
• Similarly, for any a in F other than 0, there exists an element b in F, such that a * b = 1.

Distributivity of multiplication over addition

• For all a, b and c in F, the following equality holds: a * (b + c) = (a * b) + (a * c).

Traits

Field	Marker trait for fields with full equivalence
FieldPartial	Marker trait for fields with partial equivalence