A field is a triple , where is a (non-empty) set with operations and which satisfy

Addition is commutative, associative, has neutral element and inverses
Multiplication is commutative, associative, has neutral element and inverses for all elements in
Addition and multiplication are distributive:

Properties

Note that a field, then, is an Abelian Group under , as well as under , alongside the distributive property for both operations.

This implies that, since A group’s identity is unique and All elements of a group have a unique inverse, and are unique, as well as its inverses under and .

Additive identity = multiplicative

Note that one can identify the ‘s of and with the field’s . Denote them as and . Then, since for any ,

it follows that . Thus, we can write it unambiguously as .

Analogously, the ‘s of and are the same as the field’s . Denote them as and . Then, since for any ,

it follows that . Thus, we can write it unambiguously as .

Given any elements , one has that

The converse is trivial; the direct implication comes from the following considerations:

If both and are equal to , then the implication holds
If , then it has an inverse , from which we can multiply on both sides to yield
Analogously if , then it has an inverse , from which we can multiply on both sides to yield

Thus, this is a direct consequence from the fact that has no multiplicative inverse.

Um Curso de Álgebra Linear, Flávio Ulhoa Coelho, Mary Lilian Lourenço. Editora EDUSP.