# Field Characteristic ## Metadata **Status**:: #x **Zettel**:: #zettel/fleeting **Created**:: [[2024-03-25]] **Topic**:: [[♯ Math]] **Tags**:: #abstract-algebra ## Synopsis The characteristic of a field is the smallest positive integer $n$ such that: $ \underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0 $ where $1$ is the multiplicative identity and $0$ is the additive identity of the field. If no such $n$ exists, the field is said to have characteristic zero. Some key points about field characteristic: - The characteristic of a field, if non-zero, is always a prime number. - A field of characteristic zero contains a subfield isomorphic to the rational numbers $\mathbb{Q}$, while a field of characteristic $p$ contains a subfield isomorphic to the integers modulo $p$, $\mathbb{Z}/p\mathbb{Z}$. - Examples of fields with characteristic zero include $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$ and the $p$-adic numbers. The finite field $GF(p^n)$ has characteristic $p$. - The characteristic can be defined equivalently as the exponent of the additive group of the field, i.e. the smallest positive $n$ such that $\underbrace{a + \cdots + a}_{n \text{ times}} = 0$ for all elements $a$ in the field[1]. So in summary, the field characteristic measures the "cyclic" behavior of the additive group and determines an important algebraic property of the field. Fields are classified as having either characteristic zero or prime characteristic. Citations: 1. [wikipedia.org](https://en.wikipedia.org/wiki/Characteristic_%28algebra%29) 2. [wolfram.com](https://mathworld.wolfram.com/FieldCharacteristic.html) ## Related to [[Elliptic Curve]] The characteristic is the order of the group by adding a point G to itself.