Field

A field is a ring like algebraic structure which, may be called a commutative division ring, as it extends the idea of a ring with identity by adding the requirements that every non zero element has a multiplicative inverse, and multiplication commutes.

Definition

A field \(\mathbb{F}\) is a set with two binary operations \(+\) and \(\times\), where:

  1. \((\mathbb{F}, +)\) is an abelian group.
  2. \((\mathbb{F} - \{0\}, \times)\) is an abelian group.
  3. The distributive laws holds: \(a \times (b + c) = a \times b + a \times c = (b + c) \times a\).

It is important to note that neither the first or the second axiom guarantee that multiplication by the additive identity is well defined. This can be recovered with the third axiom.

The fact that

\[ 0 \times a = a \times 0 = 0 \times 0 = 0\]

as is true in general rings, can be recovered by the third axiom as follows:

\[ a \times 0 = a \times (1 - 1) = a - a = 0.\]

This is precisely why the distributive law is given in both forms, as commutativity with \(0\) is not otherwise defined.

As a consequence of these nuances, there are many alternative ways of defining a field.

We can also characterise a field in terms of rings with identity simply as:

Theorem

A commutative ring with identity \(R\) is a field if and only:

\[ R^{\ast} = R - \{0\},\]

that is the multiplicative group with the addition of \(0\) is equal to the ring itself.

In this case we guarantee the second axiom from the definition, while the first and third follow from the fact that \(R\) is a commutative ring.

It is important also to mention that every field is an integral domain since every division ring is a domain.