Question

Prove that every finite integral domains is a field

Answer 1

Answer:

Every finite integral domain is a field.

Step-by-step explanation:

Theorem

Every finite integral domain is a field.

Proof  :

The only thing we need to show is that a typical element a ≠ 0 has a multiplicative inverse.

Consider a, a2, a3, ... Since there are only finitely many elements we must have am = an for some m < n(say).

Then 0 = am - an = am(1 - an-m). Since there are no zero-divisors we must have am ≠ 0 and hence 1 - an-m = 0 and so 1 = a(an-m-1) and we have found a multiplicative inverse for a.