Field in Ring Theory
Table of Contents
- A set \(F\) with two operations addition and multiplication that behave as corresponding operations in Real or Rational numbers.
- A set \(F\) with two operations addition and multiplication such that:
- \(F\) is a abelian group under addition
- \(F \backslash \{0\}\) is an abelian group under multiplication
- Multiplication distributes over addition
- A nonzero (i.e. not trivial ring) commutative ring where every nonzero element has an inverse is called a Field.
1. Finite Field
2. Independent use of the term "Field"
The following use of the term "Field" are unrelated to the term "Field" in Ring theory.
2.1. Field of Sets
A collection \(\mathcal{F}\) of subsets of set \(\Omega\) closed under complements and finite unions is called a field of sets \((\Omega, \mathcal{F})\) or an algebra over \(\Omega\).
This is different from Sigma Algebra which is closed under countable unions and not just finite unions.
\(\sigma\) algbera \(\subseteq\) Field of sets.
2.2. Field in Physics
In physics, a field is a continuous distribution of a physical quantity across space and time, influencing particles and interactions everywhere.