Definition:Multiplicative Inverse/Field

Definition

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$.

Let $a \in F$ such that $a \ne 0_F$.

Then the inverse element of $a$ with respect to the $\times$ operator is called the multiplicative inverse of $F$.

It is usually denoted $a^{-1}$ or $\dfrac 1 a$.

Also see

• Definition:Division Product
• Multiplicative Inverse in Field is Unique

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $3$: Field Theory: Definition and Examples of Field Structure: $\S 87$