Field

A field is a structure of abstract algebra, similar to a group or a ring. Informally, fields are the general structure in which the usual laws of arithmetic governing the operations $+, -, \times$ and $\div$ hold. In particular, the rational numbers $\mathbb{Q}$, the real numbers $\mathbb{R}$, and the complex numbers $\mathbb{C}$ are all fields.

Formally, a field $F$ is a set of elements with two operations, usually called multiplication and addition (denoted $\cdot$ and $+$, respectively) which have the following properties:

  • A field is a ring. Thus, a field obeys all of the ring axioms.
  • $1 \neq 0$, where 1 is the multiplicative identity and 0 is the additive indentity. Thus fields have at least 2 elements.
  • If we exclude 0, the remaining elements form an abelian group under multiplication. In particular, multiplicative inverses exist for every element other than 0.

Common examples of fields are the rational numbers $\mathbb{Q}$, the real numbers $\mathbb{R}$, or the integers $\mathbb{Z}$ taken modulo some prime $p$, denoted $\mathbb{F}_{p}$ or $\mathbb{Z}/p\mathbb{Z}$. In each case, addition and multiplication are defined "as usual." Other examples include the set of algebraic numbers and finite fields of order $p^{k}$ for $k$ an arbitrary positive integer.

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