Fields

A field is a set \(\mathbb{F}\) together with two binary operations \(+\) and \(\cdot\), which fulfills

  1. \(\mathbb{F}\), together with the operation \(+\) is an Abelian group
  2. \(\mathbb{F}\backslash\set{0_+}\)3, together with the operation \(\cdot\) is an Abelian group,
  3. \(\cdot\) is distributive with respect to \(+\). This means that: \[\begin{align} \forall a, b, c \in \mathbb{F}: &a \cdot (b + c) = a \cdot b + a \cdot c \\ & (a + b) \cdot c = a \cdot c + b \cdot c \end{align}\]

Some important examples of fields include the field of the real numbers without \(0\) (\(\mathbb{R}_0\)) together with multiplication and addition, and the field of the complex numbers without \(0\) (\(\mathbb{C}_0\)) with the operations addition and multiplication.

In some sense, this distributive property just means that scalar multiplication is a bilinear operation.

  1. The set \(\mathbb{F}\) without the identity element of the operation \(+\).}↩︎