Definition:Field (Abstract Algebra)/Addition

Definition

The distributand $+$ of a field $\struct {F, +, \times}$ is referred to as field addition, or just addition.

Additive Group

The group $\struct {F, +}$ is known as the additive group of $F$.

Additive Inverse

Let $\struct {F, +, \times}$ be a field whose addition operation is $+$.

Let $a \in R$ be any arbitrary element of $F$.

The additive inverse of $a$ is its inverse under addition, denoted $-a$:

$a + \paren {-a} = 0_F$

where $0_F$ is the zero of $R$.

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $3$: Field Theory: Definition and Examples of Field Structure: $\S 87$