# A Finite Integral Domain is a Field: Proof

A finite integral domain is a field. If R is a finite integral domain, then it must be a field. In this article, we will prove that every finite integral domain is a field.

## Every finite integral domain is field

Let R be a finite integral domain with n elements 1, a1, a2, a3, …, an-1, 1 being the unity in R. We will prove that R is a field.

That is, we will show that every non-zero element of R is a unit. Let ar ∈ R be a non-zero element.

Let us consider the products ar⋅1, ar⋅a1, ar⋅a2, ar⋅a3, …, ar⋅an-1. All these elements belong to R. As R is an integral domain, it does not contain any zero divisors. This forces that none of these products will be zero.

This proves our claim. As ar⋅a1, ar⋅a2, ar⋅a3, …, ar⋅an-1 are (n-1) distinct non-zero elements of R, one of them must be unity.

⇒ ar⋅as = 1 for some 1 ≤ s ≤ n-1.

⇒ ar⋅as = as⋅ar = 1, since R is a commutative ring.

This shows that ar is a unit.

Since ar is an arbitrary non-zero element of R, and it is a unit, we deduce that each non-zero element of R is a unit. Therefore, R is a field. This completes the proof of the fact that each finite integral domain is a field.

Related Topics: Introduction to Ring Theory

Units of a Ring

Characteristic of a Ring

Zero divisors of a ring

Idempotent and Nilpotent Elements

## FAQs

Q1: Every finite integral domain is a field but converse is not true.

Answer: Yes, every finite integral domain is a field, but the converse is not true. For example, (ℝ, +, ⋅) is a field, but not a finite integral domain.

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