A nonzero algebra D over a field in which, for every element a and every nonzero element b, there exists precisely one element x with bx = a.
Division algebras are not assumed to be commutative or associative.
Over the reals the associative examples are the reals (dimension 1), complex numbers (2), and quaternions (4); by Wedderburn's little theorem, every finite division algebra is a finite field.