## Integral Domains

An integral domain is a commutative ring with no zero-divisors: or Examples

1. The ring is an integral domain. (This explains the name.)

2. The polynomial rings and are integral domains.
(Look at the degree of a polynomial to see how to prove this.)

3. The ring is an integral domain.

4. If is prime, the ring is an integral domain.

1 and 3 follow from the properties of real numbers.

2 follows from the linear independence of distinct powers of If and then where If then the coefficient of for each so we have linearly independent equations each equal 0 and unknowns (if we treat the as unknown and solve them in terms of the ) or unknown (if we treat the as unknown and solve them in terms of the In either case there are more equations than unknowns so each unknown coefficient is equal to zero.

4 follows from the fact that or by Euclid's Lemma hence or If are elements of a field with then if it has an inverse and  multiplying both sides by this gives Hence there are no zero-divisors and we have that every field is an integral domain.

If is an integral domain then cancellation for multiplication is permissible. If and then This follows from that exists, so  