# Category:Division Algebras

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This category contains results about **Division Algebras**.

Definitions specific to this category can be found in Definitions/Division Algebras.

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

### Definition 1

$\struct {A_F, \oplus}$ is a **division algebra** if and only if:

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

### Definition 2

$\struct {A_F, \oplus}$ is a **division algebra** if and only if it has no zero divisors:

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

## Subcategories

This category has the following 2 subcategories, out of 2 total.

### N

- Normed Division Algebras (4 P)

### U

## Pages in category "Division Algebras"

The following 4 pages are in this category, out of 4 total.