Field

A field is a commutative ring in which every nonzero element has a multiplicative inverse. Equivalently, a field is a set with addition and multiplication such that both $(F, +, 0)$ and $(F \setminus \{0\}, \cdot, 1)$ are abelian groups, and multiplication distributes over addition.

The rationals $\mathbb{Q}$, the reals $\mathbb{R}$, and the complex numbers $\mathbb{C}$ are fields. For each prime $p$, the integers modulo $p$ form a finite field $\mathbb{F}_p$.

Fields are the algebraic structures where full arithmetic — addition, subtraction, multiplication, and division — is available. They serve as the scalars for vector spaces and underpin linear algebra.