Field theory in Mathematics is an important topic where we study sets equipped with two operations + and ×. In this article, we will study fields in abstract algebra with its definition, examples, and properties.

Table of Contents

## Definition

A non-trivial commutative ring with unity is called a field if its every non-zero elements is a unit.

More specifically, a non-empty set F equipped with two binary operations + and × forms a field if the following conditions hold.

- (F, +) is a commutative group.
- (F, ×) is a semigroup in which the multiplicative identity exists.
- Distributive laws hold on F.
- a×b = b×a for all a, b in F.

## Examples

A few examples of fields are listed below.

1. (ℝ, +, ⋅), (ℚ, +, ⋅) are commutative rings with unity 1 in which every non-zero element is a unit, so they are fields.

2. Let ℚ[√2] = {a+b√2 : a, b ∈ ℚ}. Then the ring (ℚ[√2], +, ⋅) is a field.

**Proof:**

Note that ℚ[√2] is a commutative ring with unity 1. Let a+b√2 ∈ ℚ[√2] be a non-zero element. So (a, b) ≠ (0, 0). Its multiplicative inverse is given by

$\dfrac{a}{a^2-2b^2} + \dfrac{-b}{a^2-2b^2} \sqrt{2}$ ∈ ℚ[√2]. |

This is because a^{2}-2b^{2} ≠ 0 and both the elements a/(a^{2}-2b^{2}) and -b/(a^{2}-2b^{2}) are rational numbers. This proves that each non-zero element is a unit. Hence ℚ[√2] is a field.

3. The ring (ℤ_{p}, +, ⋅) of integers modulo p is a field where p is a prime number. This is a finite field of p elements.

## Properties

The properties of a field are listed below.

- A field is an integral domain, but the converse is not true always. For example, (ℤ, +, ⋅) is an integral domain but it is not a field.
- The non-zero elements of a field F form a commutative group under multiplication, i.e., (F
^{×}, ⋅) is a commutative group, where F^{×}= F-{0}. - A finite integral domain is field. For a proof, see here.
- A finite non-trivial commutative ring with no zero divisors is a field.
- A finite division ring is a field.
- The characteristic of a field is either zero or a prime number. This follows from the fact that every field is an integral domain D and char D = 0 or a prime number always.

**Main Topic:** Introduction to Ring Theory

Prove that Every Finite Integral Domain is a Field

Idempotent and Nilpotent Elements

## FAQs

**Q1: What is a field in Mathematics?**

Answer: A field in Mathematics is a non-trivial commutative ring with unity in which every non-zero element is a unit. For example, the ring (ℝ, +, ⋅) is a field, but (ℤ, +, ⋅) is not.