# Definition:Algebra over Field

## Definition

Let $F$ be a field.

An **algebra over $F$** is an ordered pair $\struct {A, *}$ where:

- $A$ is a vector space over $F$
- $* : A^2 \to A$ is a bilinear mapping

That is, it is an algebra $\struct {A, *}$ over the ring $F$ where:

- $F$ is a field
- the $F$-module $A$ is a vector space.

The symbol $A$ is often used for such an algebra, more so as the level of abstraction increases.

### Multiplication

The bilinear mapping $*$ is often referred to as **multiplication**.

## Also defined as

Some sources insist that an **algebra over a field** requires that the bilinear mapping $*$ must have an identity element $1_A$ such that:

- $\forall a \in A: a * 1_A = 1_A * a = a$

that is, that $\struct {A, *}$ has to be a unitary algebra.

It is worth being certain of what is meant in any works read.

Especially in commutative algebra, an **algebra over a field** is often defined as a **unital associative commutative algebra**.

## Examples

### Vectors in $3$-Space with Cross Product

Let $V$ be the vector space formed of the set of all vectors in space.

Then $\struct {V, \times}$ forms an algebra over the field of vectors in space where $\times$ is the vector cross product.

## Also known as

Some sources refer to an **algebra over a field** as a **linear algebra**.

Others call it a **hypercomplex number system**.

## Also see

- Results about
**algebras over fields**can be found**here**.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**algebra over field** - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem - 2002: John C. Baez:
*The Octonions: 1.1 Preliminaries* - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**algebra**:**2.**