Let $K$ be a [[subset]] of the [[Complex number|complex numbers]] $\mathbb C$. $K$ is a **field** if it satisfies the following conditions: - (a) If $x, y$ are elements of $K$, then $x+y$ and $x y$ are also elements of $K$. - ([[Closure]] of [[addition]] and [[multiplication]]) - (b) If $x \in K$, then $-x$ is also an element of $K$. If furthermore $x \neq 0$, then $x^{-1}$ is an element of $K$. - (Existence of [[additive inverse]] for all elements) - (Existence of [[multiplicative inverse]] for non-zero elements) - (c) The elements 0 and 1 are elements of $K$. - (Existence of [[additive identity]] and [[multiplicative identity]]) The [[Element of a Set|elements]] of fields are called [[number|numbers]] Subsets of a field that are also fields are called [[subfield|subfields]] Examples: $\mathbb{C,R,Q}$ are fields