Question

Every finite integral domain is a field​

Answer 1

To prove :

A finite integral domain is a field .

Proof :

Let D be a finite integral domain with unity 1 . Let a be any non-zero element of D , then we must show that a is a unit .

If a = 1 , a is its own inverse , so we may assume that a ≠ 1 .

Now , consider the following sequence of elements of D : a , a² , a³ , . . .

Since D is finite , there must be two positive integers m and n such that m > n and aᵐ = aⁿ .

Then by cancellation , aᵐ⁻ⁿ = 1 .

Since a ≠ 1 , m - n > 1 .

Now ,

Let b ≠ 0 the inverse of a , then ab = ba = 1 .

→ ab = 1

→ ab = aᵐ⁻ⁿ (°•° aᵐ⁻ⁿ = 1)

→ b = aᵐ⁻ⁿ⁻¹

Thus , we have shown that aᵐ⁻ⁿ⁻¹ is inverse of a .

Hence , D is a field .