Definition:Field (Abstract Algebra)/Definition 3

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Definition

Let $\struct {F, +, \times}$ be an algebraic structure.


Let $\struct {F, +, \times}$ be a commutative ring with unity $\struct {F, +, \times}$ such that every non-zero element $a$ of $F$ has a multiplicative inverse:

$a^{-1}$ such that $a \times a^{-1} = 1_F = a^{-1} \times a$

where $1_F$ denotes the unity of $\struct {F, +, \times}$.


Then $\struct {F, +, \times}$ is a field.


Also defined as

Some sources do not insist that the field product of a field is commutative.

That is, what they define as a field, $\mathsf{Pr} \infty \mathsf{fWiki}$ defines as a division ring.

When they wish to refer to a field in which the field product is commutative, the term commutative field is used.


Examples

Field of rational numbers

The field of rational numbers $\struct {\Q, + \times, \le}$ is the set of rational numbers under the two operations of addition and multiplication, with an ordering $\le$ compatible with the ring structure of $\Q$.


Field of real numbers

The field of real numbers $\struct {\R, +, \times, \le}$ is the set of real numbers under the two operations of addition and multiplication, with an ordering $\le$ compatible with the ring structure of $\R$..


Field of complex numbers

The field of complex numbers $\struct {\C, +, \times}$ is the set of complex numbers under the two operations of addition and multiplication.


Field of integers modulo $p$

Let $p \in \Bbb P$ be a prime number.

Let $\Z_p$ be the set of integers modulo $p$.

Let $+_p$ and $\times_p$ denote addition modulo $p$ and multiplication modulo $p$ respectively.


The algebraic structure $\struct {\Z_p, +_p, \times_p}$ is the field of integers modulo $p$.


Smallest fields

The smallest field is the set of integers modulo $2$ under modulo addition and modulo multiplication:

$\struct {\Z_2, +_2, \times_2}$

This field has $2$ elements.


Also see


Sources

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