Fields

A Field is a tuple \((\mathbb{F}, +, \times, 0,1)\), where \(\mathbb{F}\) is the set of elements, \(+\) is the addition operator, \(\times\) is the multiplication operator, \(0\in \mathbb{F}\) is the additive identity, and \(1\in \mathbb{F}\) is the multiplicative identity. \(\mathbb{F}\) must have the following properties:

1. \((\mathbb{F}, +, 0)\) forms a group.
2. \(\times\) is associative.
3. \(\times\) distributes over \(+\).
4. \((\mathbb{F} \setminus \{0\}, \times, 1)\) forms a group.