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 a2-2b2 ≠ 0 and both the elements a/(a2-2b2) and -b/(a2-2b2) are rational numbers. This proves that each non-zero element is a unit. Hence ℚ[√2] is a field.
3. The ring (ℤp, +, ⋅) 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
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.
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