# Introduction

Given a quadratic form $Q$ defined on a vector space $V$ over a field $F$, the Clifford algebra of $Q$ is an associative $F$-algebra $C$ with a vector space homomorphism $f : V\to C$ such that $f(v)^2 = Q(v)$ for all $v\in V$. Furthermore, the triple $(C,V,f)$ has the universal property that if $A$ is any associative algebra with a homomorphism $g : V \to A$ such that $g(v)^2 = Q(v)$ for all $v\in V$, then there is a unique algebra homomorphism $h : C \to A$ such that $hf = g$. It can be shown that $f$ is injective and therefore we may identify $V$ with its image in $C$. If the dimension of $V$ is $n$, then the dimension of $C$ is $2^n$. We shall refer to $V$ as the *quadratic space* of $C$.

The primary references for quadratic forms and Clifford algebras are [[Chevalley, 1997](../../references.md#cite-chev-1954)] and [[Artin, 1957](../../references.md#cite-artin-1957)]. A more recent account, with applications is [[Lounesto, 2001](../../references.md#cite-lounesto-2001)].