Definition:Division Algebra

Definition

Let $\left({A_F, \oplus}\right)$ be an algebra over a field $F$ such that $A_F$ does not consist solely of the zero vector $\mathbf 0_A$ of $A_F$.

Then $\left({A_F, \oplus}\right)$ is a division algebra iff:

$\forall a, b \in A_F, b \ne \mathbf 0_A: \exists_1 x \in A_F, y \in A_F: a = b \oplus x, a = y \oplus b$

That is, for every pair of elements $a, b$ of the algebra where $b$ is non-zero, there exists:

a unique element $x$ such that $a = b \oplus x$

a unique element $y$ such that $a = y \oplus b$

Alternative Definition

$A$ is a division algebra iff it has no zero divisors:

$\forall a, b \in A_F: a \oplus b = \mathbf 0_A \implies a = \mathbf 0_A \lor b = \mathbf 0_A$

The two definitions are shown to be equivalent in Division Algebra has No Zero Divisors.

Sources

• John C. Baez: The Octonions (2002): 1.1 Preliminaries