Field

Definition

A field is a ring $F$ in which the multiplication operation $\cdot$ is commutative and every non-zero element has a multiplicative inverse. Formally, a field is a structure $(F,+,\cdot ,0,1)$ such that:

• $(F,+,0)$ is an Abelian Group.
• $(F\setminus \{0\},\cdot ,1)$ is an Abelian Group.
• The distributive law holds: $a\cdot (b+c)=(a\cdot b)+(a\cdot c)$.

In the language of Category Theory and Type Theory, a field can be viewed as a commutative ring where the non-zero elements form a group under multiplication.

Properties

• Characteristic: The characteristic of a field is either $0$ or a prime number $p$.
• Division: Because every non-zero element $a$ has an inverse $a^{-1}$, division $b/a$ is uniquely defined as $b\cdot a^{-1}$.
• Subfields: A subset of a field that is itself a field under the induced operations is called a subfield.

Examples

• Rational Numbers: The set $\mathbb{Q}$ is the smallest field containing the integers $\mathbb{Z}$.
• Real and Complex Numbers: Both $\mathbb{R}$ and $\mathbb{C}$ are fields, with $\mathbb{C}$ being the Algebraic Closure of $\mathbb{R}$.
• Finite Fields: Also known as Galois Fields, denoted ${\mathbb{F}}_p$ or $GF(p)$, these contain a finite number of elements.

Related Concepts

• Ring: The base structure for a field
• Commutative Ring: A ring with commutative multiplication
• Vector Space: A structure defined over a field
• Skew Field: A structure that satisfies field axioms except commutativity of multiplication (also called a Division Ring)